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			<h1 id="firstHeading" class="firstHeading">Laplace transform</h1>
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				<p>In <a href="http://en.wikipedia.org/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Laplace transform</b> is a widely used <a href="http://en.wikipedia.org/wiki/Integral_transform" title="Integral transform">integral transform</a>. Denoted <img class="tex" alt="\displaystyle\mathcal{L} \left\{f(t)\right\}" src="wikipedia-Laplace_transform_pliki/4e937d3f35445e704445b1cd32d5592e.png">, it is a <a href="http://en.wikipedia.org/wiki/Linear_operator" title="Linear operator" class="mw-redirect">linear operator</a> of a function <i>f</i>(<i>t</i>) with a real argument <i>t</i> (<i>t</i> ≥ 0) that transforms it to a function <i>F</i>(<i>s</i>) with a complex argument <i>s</i>. This transformation is essentially <a href="http://en.wikipedia.org/wiki/Bijection" title="Bijection">bijective</a> for the majority of practical uses; the respective pairs of <i>f</i>(<i>t</i>) and <i>F</i>(<i>s</i>)
 are matched in tables. The Laplace transform has the useful property 
that many relationships and operations over the originals <i>f</i>(<i>t</i>) correspond to simpler relationships and operations over the images <i>F</i>(<i>s</i>).<sup id="cite_ref-0" class="reference"><a href="#cite_note-0"><span>[</span>1<span>]</span></a></sup> The Laplace transform has many important applications throughout the sciences. It is named for <a href="http://en.wikipedia.org/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> who introduced the transform in his work on <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">probability theory</a>.</p>
<p>The Laplace transform is related to the <a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, but whereas the Fourier transform resolves a function or signal into its modes of <a href="http://en.wikipedia.org/wiki/Vibration" title="Vibration">vibration</a>, the Laplace transform resolves a function into its <a href="http://en.wikipedia.org/wiki/Moment_%28mathematics%29" title="Moment (mathematics)">moments</a>.
 Like the Fourier transform, the Laplace transform is used for solving 
differential and integral equations. In physics and engineering, it is 
used for analysis of <a href="http://en.wikipedia.org/wiki/LTI_system" title="LTI system" class="mw-redirect">linear time-invariant</a> <a href="http://en.wikipedia.org/wiki/Dynamic_system" title="Dynamic system" class="mw-redirect">systems</a> such as <a href="http://en.wikipedia.org/wiki/Electrical_circuit" title="Electrical circuit" class="mw-redirect">electrical circuits</a>, <a href="http://en.wikipedia.org/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillators</a>, <a href="http://en.wikipedia.org/wiki/Optical_device" title="Optical device" class="mw-redirect">optical devices</a>, and mechanical systems. In this analysis, the Laplace transform is often interpreted as a transformation from the <i><a href="http://en.wikipedia.org/wiki/Time-domain" title="Time-domain" class="mw-redirect">time-domain</a></i>, in which inputs and outputs are functions of time, to the <i><a href="http://en.wikipedia.org/wiki/Frequency-domain" title="Frequency-domain" class="mw-redirect">frequency-domain</a></i>, where the same inputs and outputs are functions of <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> <a href="http://en.wikipedia.org/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, in <a href="http://en.wikipedia.org/wiki/Radians" title="Radians" class="mw-redirect">radians</a>
 per unit time. Given a simple mathematical or functional description of
 an input or output to a system, the Laplace transform provides an 
alternative functional description that often simplifies the process of 
analyzing the behavior of the system, or in synthesizing a new system 
based on a set of specifications.</p>
<table id="toc" class="toc">
<tbody><tr>
<td>
<div id="toctitle">
<h2>Contents</h2>
 <span class="toctoggle">[<a href="#" class="internal" id="togglelink">hide</a>]</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#History"><span class="tocnumber">1</span> <span class="toctext">History</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Formal_definition"><span class="tocnumber">2</span> <span class="toctext">Formal definition</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Probability_theory"><span class="tocnumber">2.1</span> <span class="toctext">Probability theory</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Bilateral_Laplace_transform"><span class="tocnumber">2.2</span> <span class="toctext">Bilateral Laplace transform</span></a></li>
<li class="toclevel-2 tocsection-5"><a href="#Inverse_Laplace_transform"><span class="tocnumber">2.3</span> <span class="toctext">Inverse Laplace transform</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-6"><a href="#Region_of_convergence"><span class="tocnumber">3</span> <span class="toctext">Region of convergence</span></a></li>
<li class="toclevel-1 tocsection-7"><a href="#Properties_and_theorems"><span class="tocnumber">4</span> <span class="toctext">Properties and theorems</span></a>
<ul>
<li class="toclevel-2 tocsection-8"><a href="#Proof_of_the_Laplace_transform_of_a_function.27s_derivative"><span class="tocnumber">4.1</span> <span class="toctext">Proof of the Laplace transform of a function's derivative</span></a></li>
<li class="toclevel-2 tocsection-9"><a href="#Evaluating_improper_integrals"><span class="tocnumber">4.2</span> <span class="toctext">Evaluating improper integrals</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Relationship_to_other_transforms"><span class="tocnumber">4.3</span> <span class="toctext">Relationship to other transforms</span></a>
<ul>
<li class="toclevel-3 tocsection-11"><a href="#Laplace.E2.80.93Stieltjes_transform"><span class="tocnumber">4.3.1</span> <span class="toctext">Laplace–Stieltjes transform</span></a></li>
<li class="toclevel-3 tocsection-12"><a href="#Fourier_transform"><span class="tocnumber">4.3.2</span> <span class="toctext">Fourier transform</span></a></li>
<li class="toclevel-3 tocsection-13"><a href="#Mellin_transform"><span class="tocnumber">4.3.3</span> <span class="toctext">Mellin transform</span></a></li>
<li class="toclevel-3 tocsection-14"><a href="#Z-transform"><span class="tocnumber">4.3.4</span> <span class="toctext">Z-transform</span></a></li>
<li class="toclevel-3 tocsection-15"><a href="#Borel_transform"><span class="tocnumber">4.3.5</span> <span class="toctext">Borel transform</span></a></li>
<li class="toclevel-3 tocsection-16"><a href="#Fundamental_relationships"><span class="tocnumber">4.3.6</span> <span class="toctext">Fundamental relationships</span></a></li>
</ul>
</li>
</ul>
</li>
<li class="toclevel-1 tocsection-17"><a href="#Table_of_selected_Laplace_transforms"><span class="tocnumber">5</span> <span class="toctext">Table of selected Laplace transforms</span></a></li>
<li class="toclevel-1 tocsection-18"><a href="#s-Domain_equivalent_circuits_and_impedances"><span class="tocnumber">6</span> <span class="toctext">s-Domain equivalent circuits and impedances</span></a></li>
<li class="toclevel-1 tocsection-19"><a href="#Examples:_How_to_apply_the_properties_and_theorems"><span class="tocnumber">7</span> <span class="toctext">Examples: How to apply the properties and theorems</span></a>
<ul>
<li class="toclevel-2 tocsection-20"><a href="#Example_1:_Solving_a_differential_equation"><span class="tocnumber">7.1</span> <span class="toctext">Example 1: Solving a differential equation</span></a></li>
<li class="toclevel-2 tocsection-21"><a href="#Example_2:_Deriving_the_complex_impedance_for_a_capacitor"><span class="tocnumber">7.2</span> <span class="toctext">Example 2: Deriving the complex impedance for a capacitor</span></a></li>
<li class="toclevel-2 tocsection-22"><a href="#Example_3:_Method_of_partial_fraction_expansion"><span class="tocnumber">7.3</span> <span class="toctext">Example 3: Method of partial fraction expansion</span></a></li>
<li class="toclevel-2 tocsection-23"><a href="#Example_4:_Mixing_sines.2C_cosines.2C_and_exponentials"><span class="tocnumber">7.4</span> <span class="toctext">Example 4: Mixing sines, cosines, and exponentials</span></a></li>
<li class="toclevel-2 tocsection-24"><a href="#Example_5:_Phase_delay"><span class="tocnumber">7.5</span> <span class="toctext">Example 5: Phase delay</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-25"><a href="#See_also"><span class="tocnumber">8</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-26"><a href="#Notes"><span class="tocnumber">9</span> <span class="toctext">Notes</span></a></li>
<li class="toclevel-1 tocsection-27"><a href="#References"><span class="tocnumber">10</span> <span class="toctext">References</span></a>
<ul>
<li class="toclevel-2 tocsection-28"><a href="#Modern"><span class="tocnumber">10.1</span> <span class="toctext">Modern</span></a></li>
<li class="toclevel-2 tocsection-29"><a href="#Historical"><span class="tocnumber">10.2</span> <span class="toctext">Historical</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-30"><a href="#External_links"><span class="tocnumber">11</span> <span class="toctext">External links</span></a></li>
</ul>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=1" title="Edit section: History">edit</a>]</span> <span class="mw-headline" id="History">History</span></h2>
<p>The Laplace transform is named in honor of <a href="http://en.wikipedia.org/wiki/Mathematician" title="Mathematician">mathematician</a> and <a href="http://en.wikipedia.org/wiki/Astronomer" title="Astronomer">astronomer</a> <a href="http://en.wikipedia.org/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a>, who used the transform in his work on <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">probability theory</a>. From 1744, <a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> investigated integrals of the form</p>
<dl>
<dd><img class="tex" alt=" z = \int X(x) e^{ax}\, dx \quad\text{ and }\quad z = \int X(x) x^A \, dx" src="wikipedia-Laplace_transform_pliki/d7979e79b778b043c46b5d672c262c4a.png"></dd>
</dl>
<p>as solutions of differential equations but did not pursue the matter very far.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span>[</span>2<span>]</span></a></sup> <a href="http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Joseph Louis Lagrange</a> was an admirer of Euler and, in his work on integrating <a href="http://en.wikipedia.org/wiki/Probability_density_function" title="Probability density function">probability density functions</a>, investigated expressions of the form</p>
<dl>
<dd><img class="tex" alt=" \int X(x) e^{- a x } a^x\, dx," src="wikipedia-Laplace_transform_pliki/532b663e848cc7c11f2618bdb67f3dd3.png"></dd>
</dl>
<p>which some modern historians have interpreted within modern Laplace transform theory.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span>[</span>3<span>]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span>[</span>4<span>]</span></a></sup><sup class="noprint Inline-Template" title="The text in the vicinity of this tag needs clarification or removal of jargon from May 2010" style="white-space:nowrap;">[<i><a href="http://en.wikipedia.org/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a></i>]</sup></p>
<p>These types of integrals seem first to have attracted Laplace's 
attention in 1782 where he was following in the spirit of Euler in using
 the integrals themselves as solutions of equations.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span>[</span>5<span>]</span></a></sup>
 However, in 1785, Laplace took the critical step forward when, rather 
than just looking for a solution in the form of an integral, he started 
to apply the transforms in the sense that was later to become popular. 
He used an integral of the form:</p>
<dl>
<dd><img class="tex" alt=" \int x^s \phi (x)\, dx," src="wikipedia-Laplace_transform_pliki/9308323a28436d0cd6e58e607b6954aa.png"></dd>
</dl>
<p>akin to a <a href="http://en.wikipedia.org/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a>, to transform the whole of a <a href="http://en.wikipedia.org/wiki/Difference_equation" title="Difference equation" class="mw-redirect">difference equation</a>,
 in order to look for solutions of the transformed equation. He then 
went on to apply the Laplace transform in the same way and started to 
derive some of its properties, beginning to appreciate its potential 
power.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span>[</span>6<span>]</span></a></sup></p>
<p>Laplace also recognised that <a href="http://en.wikipedia.org/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>'s method of <a href="http://en.wikipedia.org/wiki/Fourier_series" title="Fourier series">Fourier series</a> for solving the <a href="http://en.wikipedia.org/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a>
 could only apply to a limited region of space as the solutions were 
periodic. In 1809, Laplace applied his transform to find solutions that 
diffused indefinitely in space.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span>[</span>7<span>]</span></a></sup></p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=2" title="Edit section: Formal definition">edit</a>]</span> <span class="mw-headline" id="Formal_definition">Formal definition</span></h2>
<p>The Laplace transform of a <a href="http://en.wikipedia.org/wiki/Function_%28mathematics%29" title="Function (mathematics)">function</a> <i>f</i>(<i>t</i>), defined for all <a href="http://en.wikipedia.org/wiki/Real_number" title="Real number">real numbers</a> <i>t</i> ≥ 0, is the function <i>F</i>(<i>s</i>), defined by:</p>
<dl>
<dd><img class="tex" alt="F(s) = \mathcal{L} \left\{f(t)\right\}=\int_0^{\infty} e^{-st} f(t) \,dt. " src="wikipedia-Laplace_transform_pliki/291f841863bc17cdcbb7fc75d0a2ec14.png"></dd>
</dl>
<p>The parameter <i>s</i> is a <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex number</a>:</p>
<dl>
<dd><img class="tex" alt="s = \sigma + i \omega, \, " src="wikipedia-Laplace_transform_pliki/1ae3de8f169c70efa9f3d47a8cda6289.png"> with real numbers σ and ω.</dd>
</dl>
<p>The meaning of the integral depends on types of functions of 
interest. A necessary condition for existence of the integral is that <i>ƒ</i> must be <a href="http://en.wikipedia.org/wiki/Locally_integrable" title="Locally integrable" class="mw-redirect">locally integrable</a> on [0,∞). For locally integrable functions that decay at infinity or are of <a href="http://en.wikipedia.org/wiki/Exponential_type" title="Exponential type">exponential type</a>, the integral can be understood as a (proper) <a href="http://en.wikipedia.org/wiki/Lebesgue_integral" title="Lebesgue integral" class="mw-redirect">Lebesgue integral</a>. However, for many applications it is necessary to regard it as a <a href="http://en.wikipedia.org/wiki/Conditionally_convergent" title="Conditionally convergent" class="mw-redirect">conditionally convergent</a> <a href="http://en.wikipedia.org/wiki/Improper_integral" title="Improper integral">improper integral</a> at ∞. Still more generally, the integral can be understood in a <a href="http://en.wikipedia.org/wiki/Distribution_%28mathematics%29" title="Distribution (mathematics)">weak sense</a>, and this is dealt with below.</p>
<p>One can define the Laplace transform of a finite <a href="http://en.wikipedia.org/wiki/Borel_measure" title="Borel measure">Borel measure</a> μ by the <a href="http://en.wikipedia.org/wiki/Lebesgue_integral" title="Lebesgue integral" class="mw-redirect">Lebesgue integral</a><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span>[</span>8<span>]</span></a></sup></p>
<dl>
<dd><img class="tex" alt="(\mathcal{L}\mu)(s) = \int_{[0,\infty)} e^{-st}d\mu(t)." src="wikipedia-Laplace_transform_pliki/51f4af7ed4c69a471ef0cc11c27afa2a.png"></dd>
</dl>
<p>An important special case is where μ is a <a href="http://en.wikipedia.org/wiki/Probability_measure" title="Probability measure">probability measure</a> or, even more specifically, the Dirac delta function. In <a href="http://en.wikipedia.org/wiki/Operational_calculus" title="Operational calculus">operational calculus</a>, the Laplace transform of a measure is often treated as though the measure came from a <a href="http://en.wikipedia.org/wiki/Distribution_function" title="Distribution function">distribution function</a> <i>ƒ</i>. In that case, to avoid potential confusion, one often writes</p>
<dl>
<dd><img class="tex" alt="(\mathcal{L}f)(s) = \int_{0^-}^\infty e^{-st}f(t)\,dt" src="wikipedia-Laplace_transform_pliki/2877ef66a2beca6d1a11322ae2c6720b.png"></dd>
</dl>
<p>where the lower limit of 0<sup>−</sup> is short notation to mean</p>
<dl>
<dd><img class="tex" alt="\lim_{\varepsilon\to 0^+}\int_{-\varepsilon}^\infty." src="wikipedia-Laplace_transform_pliki/b548a8135a067ccbd96b5eab16f7e0f2.png"></dd>
</dl>
<p>This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the <a href="http://en.wikipedia.org/wiki/Lebesgue_integral" title="Lebesgue integral" class="mw-redirect">Lebesgue integral</a>, it is not necessary to take such a limit, it does appear more naturally in connection with the <a href="http://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform" title="Laplace–Stieltjes transform">Laplace–Stieltjes transform</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=3" title="Edit section: Probability theory">edit</a>]</span> <span class="mw-headline" id="Probability_theory">Probability theory</span></h3>
<p>In <a href="http://en.wikipedia.org/wiki/Probability_theory" title="Probability theory">pure</a> and <a href="http://en.wikipedia.org/wiki/Applied_probability" title="Applied probability">applied probability</a>, the Laplace transform is defined by means of an <a href="http://en.wikipedia.org/wiki/Expectation_value" title="Expectation value" class="mw-redirect">expectation value</a>. If <i>X</i> is a <a href="http://en.wikipedia.org/wiki/Random_variable" title="Random variable">random variable</a> with <a href="http://en.wikipedia.org/wiki/Probability_density_function" title="Probability density function">probability density function</a> <i>ƒ</i>, then the Laplace transform of <i>ƒ</i> is given by the expectation</p>
<dl>
<dd><img class="tex" alt="(\mathcal{L}f)(s) = E\left[e^{-sX} \right]. \, " src="wikipedia-Laplace_transform_pliki/29876b6fa185b11d8628189a7f3840b1.png"></dd>
</dl>
<p>By <a href="http://en.wikipedia.org/wiki/Abuse_of_notation" title="Abuse of notation">abuse of language</a>, this is referred to as the Laplace transform of the random variable <i>X</i> itself. Replacing <i>s</i> by −<i>t</i> gives the <a href="http://en.wikipedia.org/wiki/Moment_generating_function" title="Moment generating function" class="mw-redirect">moment generating function</a> of <i>X</i>. The Laplace transform has applications throughout probability theory, including <a href="http://en.wikipedia.org/wiki/First_passage_time" title="First passage time" class="mw-redirect">first passage times</a> of <a href="http://en.wikipedia.org/wiki/Stochastic_processes" title="Stochastic processes" class="mw-redirect">stochastic processes</a> such as <a href="http://en.wikipedia.org/wiki/Markov_chain" title="Markov chain">Markov chains</a>, and <a href="http://en.wikipedia.org/wiki/Renewal_theory" title="Renewal theory">renewal theory</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=4" title="Edit section: Bilateral Laplace transform">edit</a>]</span> <span class="mw-headline" id="Bilateral_Laplace_transform">Bilateral Laplace transform</span></h3>
<div class="rellink relarticle mainarticle">Main article: <a href="http://en.wikipedia.org/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">Two-sided Laplace transform</a></div>
<p>When one says "the Laplace transform" without qualification, the 
unilateral or one-sided transform is normally intended. The Laplace 
transform can be alternatively defined as the <i>bilateral Laplace transform</i> or <a href="http://en.wikipedia.org/wiki/Two-sided_Laplace_transform" title="Two-sided Laplace transform">two-sided Laplace transform</a>
 by extending the limits of integration to be the entire real axis. If 
that is done the common unilateral transform simply becomes a special 
case of the bilateral transform where the definition of the function 
being transformed is multiplied by the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>.</p>
<p>The bilateral Laplace transform is defined as follows:</p>
<dl>
<dd><img class="tex" alt="F(s)  = \mathcal{L}\left\{f(t)\right\}  =\int_{-\infty}^{\infty} e^{-st} f(t)\,dt." src="wikipedia-Laplace_transform_pliki/f2a613fc61132e4b8f053ed85030a651.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=5" title="Edit section: Inverse Laplace transform">edit</a>]</span> <span class="mw-headline" id="Inverse_Laplace_transform">Inverse Laplace transform</span></h3>
<div class="rellink boilerplate seealso">For more details on this topic, see <a href="http://en.wikipedia.org/wiki/Inverse_Laplace_transform" title="Inverse Laplace transform">Inverse Laplace transform</a>.</div>
<p>The <a href="http://en.wikipedia.org/wiki/Inverse_Laplace_transform" title="Inverse Laplace transform">inverse Laplace transform</a> is given by the following <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> integral, which is known by various names (the <b>Bromwich integral</b>, the <b>Fourier-Mellin integral</b>, and <b>Mellin's inverse formula</b>):</p>
<dl>
<dd><img class="tex" alt="f(t) = \mathcal{L}^{-1} \{F(s)\} = \frac{1}{2 \pi i} \lim_{T\to\infty}\int_{ \gamma - i T}^{ \gamma + i T} e^{st} F(s)\,ds," src="wikipedia-Laplace_transform_pliki/a9e6454f0de3496620bdf86400ad3d52.png"></dd>
</dl>
<p>where <span class="texhtml">γ</span> is a real number so that the contour path of integration is in the <i><a href="http://en.wikipedia.org/wiki/Region_of_convergence" title="Region of convergence" class="mw-redirect">region of convergence</a></i> of <i>F</i>(<i>s</i>). An alternative formula for the inverse Laplace transform is given by <a href="http://en.wikipedia.org/wiki/Post%27s_inversion_formula" title="Post's inversion formula">Post's inversion formula</a>.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=6" title="Edit section: Region of convergence">edit</a>]</span> <span class="mw-headline" id="Region_of_convergence">Region of convergence</span></h2>
<p>If <i>ƒ</i> is a <a href="http://en.wikipedia.org/wiki/Locally_integrable" title="Locally integrable" class="mw-redirect">locally integrable</a> function (or more generally a Borel measure locally of bounded variation), then the Laplace transform <i>F</i>(<i>s</i>) of <i>ƒ</i> converges provided that the limit</p>
<dl>
<dd><img class="tex" alt="\lim_{R\to\infty}\int_0^R f(t)e^{-ts}\,dt" src="wikipedia-Laplace_transform_pliki/f8285f93d71f33d026a7474682d9f961.png"></dd>
</dl>
<p>exists. The Laplace transform converges absolutely if the integral</p>
<dl>
<dd><img class="tex" alt="\int_0^\infty |f(t)e^{-ts}|\,dt" src="wikipedia-Laplace_transform_pliki/e7a0cd869dd785c7d53dff040aa4d2df.png"></dd>
</dl>
<p>exists (as proper Lebesgue integral). The Laplace transform is 
usually understood as conditionally convergent, meaning that it 
converges in the former instead of the latter sense.</p>
<p>The set of values for which <i>F</i>(<i>s</i>) converges absolutely is either of the form Re{<i>s</i>} &gt; <i>a</i> or else Re{<i>s</i>} ≥ <i>a</i>, where <i>a</i> is an <a href="http://en.wikipedia.org/wiki/Extended_real_number" title="Extended real number" class="mw-redirect">extended real constant</a>, −∞&nbsp;≤&nbsp;<i>a</i>&nbsp;≤&nbsp;∞. (This follows from the <a href="http://en.wikipedia.org/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">dominated convergence theorem</a>.) The constant <i>a</i> is known as the abscissa of absolute convergence, and depends on the growth behavior of <i>ƒ</i>(<i>t</i>).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span>[</span>9<span>]</span></a></sup> Analogously, the two-sided transform converges absolutely in a strip of the form <i>a</i> &lt; Re{<i>s</i>} &lt; <i>b</i>, and possibly including the lines Re{<i>s</i>}&nbsp;=&nbsp;<i>a</i> or Re{<i>s</i>}&nbsp;=&nbsp;<i>b</i>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span>[</span>10<span>]</span></a></sup> The subset of values of <i>s</i>
 for which the Laplace transform converges absolutely is called the 
region of absolute convergence or the domain of absolute convergence. In
 the two-sided case, it is sometimes called the strip of absolute 
convergence. The Laplace transform is <a href="http://en.wikipedia.org/wiki/Analytic_function" title="Analytic function">analytic</a> in the region of absolute convergence.</p>
<p>Similarly, the set of values for which <i>F</i>(<i>s</i>) converges (conditionally or absolutely) is known as the region of conditional convergence, or simply the <b>region of convergence</b> (ROC). If the Laplace transform converges (conditionally) at <i>s</i>&nbsp;=&nbsp;<i>s</i><sub>0</sub>, then it automatically converges for all <i>s</i> with Re{<i>s</i>}&nbsp;&gt;&nbsp;Re{<i>s</i><sub>0</sub>}. Therefore the region of convergence is a half-plane of the form Re{<i>s</i>}&nbsp;&gt;&nbsp;<i>a</i>, possibly including some points of the boundary line Re{<i>s</i>}&nbsp;=&nbsp;<i>a</i>. In the region of convergence Re{<i>s</i>} &gt; Re{<i>s</i><sub>0</sub>}, the Laplace transform of <i>ƒ</i> can be expressed by <a href="http://en.wikipedia.org/wiki/Integration_by_parts" title="Integration by parts">integrating by parts</a> as the integral</p>
<dl>
<dd><img class="tex" alt="F(s) = (s-s_0)\int_0^\infty e^{-(s-s_0)t}\beta(t)\,dt,\quad \beta(u)=\int_0^u e^{-s_0t}f(t)\,dt." src="wikipedia-Laplace_transform_pliki/7161b548d055474e60362cea8fb7c5f9.png"></dd>
</dl>
<p>That is, in the region of convergence <i>F</i>(<i>s</i>) can 
effectively be expressed as the absolutely convergent Laplace transform 
of some other function. In particular, it is analytic.</p>
<p>A variety of theorems, in the form of <a href="http://en.wikipedia.org/wiki/Paley%E2%80%93Wiener_theorem" title="Paley–Wiener theorem">Paley–Wiener theorems</a>, exist concerning the relationship between the decay properties of <i>ƒ</i> and the properties of the Laplace transform within the region of convergence.</p>
<p>In engineering applications, a function corresponding to a <a href="http://en.wikipedia.org/wiki/LTI_system" title="LTI system" class="mw-redirect">linear time-invariant (LTI) system</a> is <i>stable</i>
 if every bounded input produces a bounded output. This is equivalent to
 the absolute convergence of the Laplace transform of the impulse 
response function in the region Re{<i>s</i>}&nbsp;≥&nbsp;0. As a result,
 LTI systems are stable provided the poles of the Laplace transform of 
the impulse response function have negative real part.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=7" title="Edit section: Properties and theorems">edit</a>]</span> <span class="mw-headline" id="Properties_and_theorems">Properties and theorems</span></h2>
<p>The Laplace transform has a number of properties that make it useful for analyzing linear <a href="http://en.wikipedia.org/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. The most significant advantage is that <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">differentiation</a> and <a href="http://en.wikipedia.org/wiki/Integral" title="Integral">integration</a> become multiplication and division, respectively, by <i>s</i> (similarly to <a href="http://en.wikipedia.org/wiki/Logarithm" title="Logarithm">logarithms</a> changing multiplication of numbers to addition of their logarithms). Because of this property, the Laplace variable <i>s</i> is also known as <i>operator variable</i> in the L domain: either <i>derivative operator</i> or (for <i>s</i><sup>−1</sup>) <i>integration operator</i>. The transform turns <a href="http://en.wikipedia.org/wiki/Integral_equation" title="Integral equation">integral equations</a> and <a href="http://en.wikipedia.org/wiki/Differential_equation" title="Differential equation">differential equations</a> to <a href="http://en.wikipedia.org/wiki/Polynomial_equation" title="Polynomial equation" class="mw-redirect">polynomial equations</a>, which are much easier to solve. Once solved, use of the inverse Laplace transform reverts back to the time domain.</p>
<p>Given the functions <i>f</i>(<i>t</i>) and <i>g</i>(<i>t</i>), and their respective Laplace transforms <i>F</i>(<i>s</i>) and <i>G</i>(<i>s</i>):</p>
<dl>
<dd><img class="tex" alt=" f(t) = \mathcal{L}^{-1} \{  F(s) \} " src="wikipedia-Laplace_transform_pliki/bfa48a9ec64283dadf4b96ec7a64efce.png"></dd>
<dd><img class="tex" alt=" g(t) = \mathcal{L}^{-1} \{  G(s) \} " src="wikipedia-Laplace_transform_pliki/e49b0518568d40dc72376680063808d2.png"></dd>
</dl>
<p>the following table is a list of properties of unilateral Laplace transform:<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span>[</span>11<span>]</span></a></sup></p>
<table class="wikitable">
<caption><b>Properties of the unilateral Laplace transform</b></caption>
<tbody><tr>
<th></th>
<th>Time domain</th>
<th>'s' domain</th>
<th>Comment</th>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Linearity" title="Linearity" class="mw-redirect">Linearity</a></th>
<td><img class="tex" alt=" a f(t) + b g(t) \ " src="wikipedia-Laplace_transform_pliki/df2830682a5971d4d87303fed7dbc0ea.png"></td>
<td><img class="tex" alt=" a F(s) + b G(s) \ " src="wikipedia-Laplace_transform_pliki/c7a80b998a228dfec3c427354ee56728.png"></td>
<td>Can be proved using basic rules of integration.</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">Frequency differentiation</a></th>
<td><img class="tex" alt=" t f(t) \ " src="wikipedia-Laplace_transform_pliki/f30be00997336bb6495ac14db76ee62d.png"></td>
<td><img class="tex" alt=" -F'(s) \ " src="wikipedia-Laplace_transform_pliki/d308ef74238fca22fb17b0bfb0ae65f8.png"></td>
<td><img class="tex" alt="F'\," src="wikipedia-Laplace_transform_pliki/cb4590eee82f4f562bba694c7c4a01cc.png"> is the first <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">derivative</a> of <img class="tex" alt="F\," src="wikipedia-Laplace_transform_pliki/bc352fc10ca296a872b51d91a1132127.png">.</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">Frequency differentiation</a></th>
<td><img class="tex" alt=" t^{n} f(t) \ " src="wikipedia-Laplace_transform_pliki/f37ddab388ee666231605ce4740d5063.png"></td>
<td><img class="tex" alt=" (-1)^{n} F^{(n)}(s) \ " src="wikipedia-Laplace_transform_pliki/f87ef8425c6ca5ee5a0a97727e1f3528.png"></td>
<td>More general form, <i>n</i><sup>th</sup> derivative of F(s).</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">Differentiation</a></th>
<td><img class="tex" alt=" f'(t) \ " src="wikipedia-Laplace_transform_pliki/79df2abee668b5848ffded8bd2426547.png"></td>
<td><img class="tex" alt=" s F(s) - f(0) \ " src="wikipedia-Laplace_transform_pliki/7109032d339052e71e4a677c804a3358.png"></td>
<td><i>ƒ</i> is assumed to be a <a href="http://en.wikipedia.org/wiki/Differentiable_function" title="Differentiable function">differentiable function</a>, and its derivative is assumed to be of <a href="http://en.wikipedia.org/wiki/Exponential_type" title="Exponential type">exponential type</a>. This can then be obtained by <a href="http://en.wikipedia.org/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a></td>
</tr>
<tr>
<th>Second <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">Differentiation</a></th>
<td><img class="tex" alt=" f''(t) \ " src="wikipedia-Laplace_transform_pliki/2d5d77d42ea885d4db40e1833b9466c3.png"></td>
<td><img class="tex" alt=" s^2 F(s) - s f(0) - f'(0) \ " src="wikipedia-Laplace_transform_pliki/85ce76769544d84a0bdba7f1b4b8b0ce.png"></td>
<td><i>ƒ</i> is assumed twice differentiable and the second derivative 
to be of exponential type. Follows by applying the Differentiation 
property to <img class="tex" alt=" f'(t)\, " src="wikipedia-Laplace_transform_pliki/a04ce07f7c97a29464aa0cf462b85131.png">.</td>
</tr>
<tr>
<th>General <a href="http://en.wikipedia.org/wiki/Derivative" title="Derivative">Differentiation</a></th>
<td><img class="tex" alt=" f^{(n)}(t)  \ " src="wikipedia-Laplace_transform_pliki/18ce4983d460e3bdf425f1d1c7ab21f1.png"></td>
<td><img class="tex" alt=" s^n F(s) - s^{n - 1} f(0) - \cdots - f^{(n - 1)}(0) \ " src="wikipedia-Laplace_transform_pliki/66000decd63c5b418fde7d54df488a8d.png"></td>
<td><i>ƒ</i> is assumed to be <i>n</i>-times differentiable, with <i>n</i><sup>th</sup> derivative of exponential type. Follow by <a href="http://en.wikipedia.org/wiki/Mathematical_induction" title="Mathematical induction">mathematical induction</a>.</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Frequency" title="Frequency">Frequency integration</a></th>
<td><img class="tex" alt=" \frac{f(t)}{t}  \ " src="wikipedia-Laplace_transform_pliki/2fb59f1e0b121394a51a0ddd956be149.png"></td>
<td><img class="tex" alt=" \int_s^\infty F(\sigma)\, d\sigma \ " src="wikipedia-Laplace_transform_pliki/f293b39c18bad22da32453f1ec372211.png"></td>
<td></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Integral" title="Integral">Integration</a></th>
<td><img class="tex" alt=" \int_0^t f(\tau)\, d\tau  =  (u * f)(t)" src="wikipedia-Laplace_transform_pliki/12d326904ebd5674e988d3ada86b61bf.png"></td>
<td><img class="tex" alt=" {1 \over s} F(s) " src="wikipedia-Laplace_transform_pliki/213cb5bab411f6d133857721df7e0d03.png"></td>
<td><span class="texhtml"><i>u</i>(<i>t</i>)</span> is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>. Note <span class="texhtml">(<i>u</i> * <i>f</i>)(<i>t</i>)</span> is the <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> of <span class="texhtml"><i>u</i>(<i>t</i>)</span> and <span class="texhtml"><i>f</i>(<i>t</i>)</span>.</td>
</tr>
<tr>
<th>Scaling</th>
<td><img class="tex" alt=" f(at) \ " src="wikipedia-Laplace_transform_pliki/d0b5548b4d441186349c40d40bee9bc1.png"></td>
<td><img class="tex" alt=" {1 \over a} F \left ( {s \over a} \right )" src="wikipedia-Laplace_transform_pliki/03204c2e34cc815422d0bd713fb2ad60.png"></td>
<td>where <span class="texhtml"><i>a</i></span> is positive.</td>
</tr>
<tr>
<th>Frequency shifting</th>
<td><img class="tex" alt=" e^{at} f(t)  \ " src="wikipedia-Laplace_transform_pliki/23038c76395b091568675a6395855b90.png"></td>
<td><img class="tex" alt=" F(s - a) \ " src="wikipedia-Laplace_transform_pliki/902a53c1c508093d91bf27b542ab79fe.png"></td>
<td></td>
</tr>
<tr>
<th>Time shifting</th>
<td><img class="tex" alt=" f(t - a) u(t - a) \ " src="wikipedia-Laplace_transform_pliki/10a00bd8ac9a55888ef7be1fe2fda387.png"></td>
<td><img class="tex" alt=" e^{-as} F(s) \ " src="wikipedia-Laplace_transform_pliki/22fc9b726798e1c37ea2a5591a820d68.png"></td>
<td><span class="texhtml"><i>u</i>(<i>t</i>)</span> is the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication">Multiplication</a></th>
<td><img class="tex" alt=" f(t) g(t) \ " src="wikipedia-Laplace_transform_pliki/a6fad331cd7e645e1a5fdce455afea9c.png"></td>
<td><img class="tex" alt=" \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}F(\sigma)G(s-\sigma)\,d\sigma \ " src="wikipedia-Laplace_transform_pliki/70393e2b0002a1a0d393bbd4acc4da9c.png"></td>
<td>the integration is done along the vertical line <span class="texhtml"><i>R</i><i>e</i>(σ) = <i>c</i></span> that lies entirely within the region of convergence of <i>F</i>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span>[</span>12<span>]</span></a></sup></td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">Convolution</a></th>
<td><img class="tex" alt=" (f * g)(t) = \int_0^t f(\tau)g(t-\tau)\,d\tau" src="wikipedia-Laplace_transform_pliki/994992a2ba70df11c5133e98e7ac794d.png"></td>
<td><img class="tex" alt=" F(s) \cdot G(s) \ " src="wikipedia-Laplace_transform_pliki/590160fde7b96a774738f37b706b8de8.png"></td>
<td><i>ƒ</i>(<i>t</i>) and <i>g</i>(<i>t</i>) are extended by zero for <i>t</i>&nbsp;&lt;&nbsp;0 in the definition of the convolution.</td>
</tr>
<tr>
<th><a href="http://en.wikipedia.org/wiki/Periodic_Function" title="Periodic Function" class="mw-redirect">Periodic Function</a></th>
<td><img class="tex" alt=" f(t) \ " src="wikipedia-Laplace_transform_pliki/5281caa9c9fe438bce47481caaa7d715.png"></td>
<td><img class="tex" alt="{1 \over 1 - e^{-Ts}} \int_0^T e^{-st} f(t)\,dt " src="wikipedia-Laplace_transform_pliki/17da807cb4d61d17fd98fcc28edd97df.png"></td>
<td><span class="texhtml"><i>f</i>(<i>t</i>)</span> is a periodic function of <a href="http://en.wikipedia.org/wiki/Periodic_function" title="Periodic function">period</a> <span class="texhtml"><i>T</i></span> so that <img class="tex" alt="f(t) = f(t + T), \; \forall t\ge 0" src="wikipedia-Laplace_transform_pliki/76fa16d845f79ea9b749f11f4499b191.png">. This is the result of the time shifting property and the <a href="http://en.wikipedia.org/wiki/Geometric_series" title="Geometric series">geometric series</a>.</td>
</tr>
</tbody></table>
<ul>
<li><b><a href="http://en.wikipedia.org/wiki/Initial_value_theorem" title="Initial value theorem">Initial value theorem</a></b>:</li>
</ul>
<dl>
<dd><img class="tex" alt="f(0^+)=\lim_{s\to \infty}{sF(s)}." src="wikipedia-Laplace_transform_pliki/b6d384ba77906a0c34d5a7f29fcc87ba.png"></dd>
</dl>
<ul>
<li><b><a href="http://en.wikipedia.org/wiki/Final_value_theorem" title="Final value theorem">Final value theorem</a></b>:</li>
</ul>
<dl>
<dd><img class="tex" alt="f(\infty)=\lim_{s\to 0}{sF(s)}" src="wikipedia-Laplace_transform_pliki/914166f18d1365289e125d6943058554.png">, if all <a href="http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29" title="Pole (complex analysis)">poles</a> of <span class="texhtml"><i>s</i><i>F</i>(<i>s</i>)</span> are in the left half-plane.</dd>
<dd>The final value theorem is useful because it gives the long-term behaviour without having to perform <a href="http://en.wikipedia.org/wiki/Partial_fraction" title="Partial fraction">partial fraction</a> decompositions or other difficult algebra. If a function's poles are in the right-hand plane (e.g. <span class="texhtml"><i>e</i><sup><i>t</i></sup></span> or <span class="texhtml">sin(<i>t</i>)</span>) the behaviour of this formula is undefined.</dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=8" title="Edit section: Proof of the Laplace transform of a function's derivative">edit</a>]</span> <span class="mw-headline" id="Proof_of_the_Laplace_transform_of_a_function.27s_derivative">Proof of the Laplace transform of a function's derivative</span></h3>
<p>It is often convenient to use the differentiation property of the 
Laplace transform to find the transform of a function's derivative. This
 can be derived from the basic expression for a Laplace transform as 
follows:</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
\mathcal{L} \left\{f(t)\right\} &amp; = \int_{0^-}^{\infty} e^{-st} f(t)\,dt \\[8pt]
&amp; = \left[\frac{f(t)e^{-st}}{-s} \right]_{0^-}^{\infty} -
\int_{0^-}^\infty \frac{e^{-st}}{-s} f'(t) \, dt\quad \text{(by parts)} \\[8pt]
&amp; = \left[-\frac{f(0)}{-s}\right] +
\frac{1}{s}\mathcal{L}\left\{f'(t)\right\},
\end{align}
" src="wikipedia-Laplace_transform_pliki/a960fcd48e314836eb535f78aabe0114.png"></dd>
</dl>
<p>yielding</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}\left\{ f'(t) \right\} = s\cdot\mathcal{L} \left\{ f(t) \right\}-f(0), " src="wikipedia-Laplace_transform_pliki/35bed0a17588d24c64bb4cdaa61e39c0.png"></dd>
</dl>
<p>and in the bilateral case,</p>
<dl>
<dd><img class="tex" alt=" \mathcal{L}\left\{ { f'(t) }  \right\}

  = s \int_{-\infty}^\infty e^{-st} f(t)\,dt  = s \cdot \mathcal{L} \{ f(t) \}. " src="wikipedia-Laplace_transform_pliki/f152d057481ea31690111c282141f45c.png"></dd>
</dl>
<p>The general result</p>
<dl>
<dd><img class="tex" alt="\mathcal{L} \left\{ f^{(n)}(t) \right\} = s^n \cdot \mathcal{L} \left\{ f(t) \right\} - s^{n - 1} f(0) - \cdots - f^{(n - 1)}(0)," src="wikipedia-Laplace_transform_pliki/6e824930a51fb5902d27432e7d810524.png"></dd>
</dl>
<p>where <i>f</i><sup><i>n</i></sup> is the <i>n</i>-th derivative of <i>f</i>, can then be established with an <a href="http://en.wikipedia.org/wiki/Mathematical_induction" title="Mathematical induction">inductive</a> argument.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=9" title="Edit section: Evaluating improper integrals">edit</a>]</span> <span class="mw-headline" id="Evaluating_improper_integrals">Evaluating improper integrals</span></h3>
<p>Let <img class="tex" alt="\mathcal{L}\left\{f(t)\right\}=F(s)" src="wikipedia-Laplace_transform_pliki/26bdefa633500c9cf0ff6a640ddb4e1b.png">, then (see the table above)</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_{s}^{\infty}F(p)\, dp," src="wikipedia-Laplace_transform_pliki/c847e0608bb511c2822d6781aa3e399e.png"></dd>
</dl>
<p>or</p>
<dl>
<dd><img class="tex" alt="\int_{0}^{\infty}\frac{f(t)}{t}e^{-st}\, dt=\int_{s}^{\infty}F(p)\, dp." src="wikipedia-Laplace_transform_pliki/1ad45ee6423dc0415c104f9732c3df43.png"></dd>
</dl>
<p>Let <img class="tex" alt="s\to 0" src="wikipedia-Laplace_transform_pliki/476374e9b651dea5d569377cb66a1fdd.png"> we get the identity</p>
<dl>
<dd><img class="tex" alt="\int_{0}^{\infty}\frac{f(t)}{t}\, dt=\int_{0}^{\infty}F(p)\, dp." src="wikipedia-Laplace_transform_pliki/65bd704e31ce27cac3b4b873d0a8cbcb.png"></dd>
</dl>
<p>For example,</p>
<dl>
<dd><img class="tex" alt="\int_{0}^{\infty}\frac{\cos at-\cos bt}{t}\, dt=\int_{0}^{\infty}\left(\frac{p}{p^{2}+a^{2}}-\frac{p}{p^{2}+b^{2}}\right)\, dp=\frac{1}{2}\left.\ln\frac{p^{2}+a^{2}}{p^{2}+b^{2}}\right|_{0}^{\infty}=\ln b-\ln a." src="wikipedia-Laplace_transform_pliki/48da902eba07bbacb88e50093f8c8eb1.png"></dd>
</dl>
<p>Another example is <a href="http://en.wikipedia.org/wiki/Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a>.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=10" title="Edit section: Relationship to other transforms">edit</a>]</span> <span class="mw-headline" id="Relationship_to_other_transforms">Relationship to other transforms</span></h3>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=11" title="Edit section: Laplace–Stieltjes transform">edit</a>]</span> <span class="mw-headline" id="Laplace.E2.80.93Stieltjes_transform">Laplace–Stieltjes transform</span></h4>
<p>The (unilateral) <a href="http://en.wikipedia.org/wiki/Laplace%E2%80%93Stieltjes_transform" title="Laplace–Stieltjes transform">Laplace–Stieltjes transform</a> of a function <i>g</i>&nbsp;:&nbsp;<b>R</b>&nbsp;→&nbsp;<b>R</b> is defined by the <a href="http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integral" title="Lebesgue–Stieltjes integral" class="mw-redirect">Lebesgue–Stieltjes integral</a></p>
<dl>
<dd><img class="tex" alt="\{\mathcal{L}^*g\}(s) = \int_0^\infty e^{-st}dg(t)." src="wikipedia-Laplace_transform_pliki/5a32c08d4d2d7358f36e488c8935452b.png"></dd>
</dl>
<p>The function <i>g</i> is assumed to be of <a href="http://en.wikipedia.org/wiki/Bounded_variation" title="Bounded variation">bounded variation</a>. If <i>g</i> is the <a href="http://en.wikipedia.org/wiki/Antiderivative" title="Antiderivative">antiderivative</a> of <i>ƒ</i>:</p>
<dl>
<dd><img class="tex" alt="g(x) = \int_0^x f(t)\,dt" src="wikipedia-Laplace_transform_pliki/4e43e2cb433486a97eb697fd64473c87.png"></dd>
</dl>
<p>then the Laplace–Stieltjes transform of <i>g</i> and the Laplace transform of <i>ƒ</i> coincide. In general, the Laplace–Stieltjes transform is the Laplace transform of the <a href="http://en.wikipedia.org/wiki/Stieltjes_measure" title="Stieltjes measure" class="mw-redirect">Stieltjes measure</a> associated to <i>g</i>.
 So in practice, the only distinction between the two transforms is that
 the Laplace transform is thought of as operating on the density 
function of the measure, whereas the Laplace–Stieltjes transform is 
thought of as operating on its <a href="http://en.wikipedia.org/wiki/Cumulative_distribution_function" title="Cumulative distribution function">cumulative distribution function</a>.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span>[</span>13<span>]</span></a></sup></p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=12" title="Edit section: Fourier transform">edit</a>]</span> <span class="mw-headline" id="Fourier_transform">Fourier transform</span></h4>
<p>The <a href="http://en.wikipedia.org/wiki/Continuous_Fourier_transform" title="Continuous Fourier transform" class="mw-redirect">continuous Fourier transform</a> is equivalent to evaluating the bilateral Laplace transform with imaginary argument <i>s</i> = <i>i</i>ω or <i>s = 2πfi</i>&nbsp;:</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
\hat{f}(\omega) &amp; = \mathcal{F}\left\{f(t)\right\} \\[1em]
&amp; = \mathcal{L}\left\{f(t)\right\}|_{s =  i\omega}  =  F(s)|_{s = i \omega}\\[1em]
&amp; = \int_{-\infty}^{\infty} e^{-\imath \omega t} f(t)\,\mathrm{d}t.\\
\end{align}
" src="wikipedia-Laplace_transform_pliki/a64f2756d3bfdca47f2130fb7c4ef72f.png"></dd>
</dl>
<p>This expression excludes the scaling factor <img class="tex" alt="1/\sqrt{2 \pi}" src="wikipedia-Laplace_transform_pliki/069b1bb488b9d6e3638d2352e8eba3b8.png">,
 which is often included in definitions of the Fourier transform. This 
relationship between the Laplace and Fourier transforms is often used to
 determine the <a href="http://en.wikipedia.org/wiki/Frequency_spectrum" title="Frequency spectrum">frequency spectrum</a> of a <a href="http://en.wikipedia.org/wiki/Signal_%28information_theory%29" title="Signal (information theory)" class="mw-redirect">signal</a> or <a href="http://en.wikipedia.org/wiki/Dynamical_system" title="Dynamical system">dynamical system</a>.</p>
<p>The above relation is valid as stated if and only if the region of convergence (ROC) of <i>F</i>(<i>s</i>) contains the imaginary axis, σ&nbsp;=&nbsp;0. For example, the function <i>f</i>(<i>t</i>)&nbsp;=&nbsp;cos(ω<sub>0</sub><i>t</i>)<i>u</i>(<i>t</i>) has a Laplace transform <i>F</i>(<i>s</i>)&nbsp;=&nbsp;<i>s</i>/(<i>s</i><sup>2</sup>&nbsp;+&nbsp;ω<sub>0</sub><sup>2</sup>) whose ROC is Re(<i>s</i>)&nbsp;&gt;&nbsp;0. Therefore, substituting <i>s</i>&nbsp;=&nbsp;<i>i</i>ω in <i>F</i>(<i>s</i>) does not yield the Fourier transform of <i>f</i>(<i>t</i>)&nbsp;=&nbsp;cos(ω<sub>0</sub><i>t</i>).</p>
<p>However, a relation of the form</p>
<dl>
<dd><img class="tex" alt="\lim_{\sigma\to 0^+} F(\sigma+i\omega) = \hat{f}(\omega)" src="wikipedia-Laplace_transform_pliki/6181efa514c0c7959bbc5e05787840c6.png"></dd>
</dl>
<p>holds under much weaker conditions. For instance, this holds for the above example provided that the limit is understood as a <a href="http://en.wikipedia.org/wiki/Weak_limit" title="Weak limit" class="mw-redirect">weak limit</a> of measures (see <a href="http://en.wikipedia.org/wiki/Vague_topology" title="Vague topology">vague topology</a>).
 General conditions relating the limit of the Laplace transform of a 
function on the boundary to the Fourier transform take the form of <a href="http://en.wikipedia.org/wiki/Paley-Wiener_theorem" title="Paley-Wiener theorem" class="mw-redirect">Paley-Wiener theorems</a>.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=13" title="Edit section: Mellin transform">edit</a>]</span> <span class="mw-headline" id="Mellin_transform">Mellin transform</span></h4>
<p>The <a href="http://en.wikipedia.org/wiki/Mellin_transform" title="Mellin transform">Mellin transform</a> and its inverse are related to the two-sided Laplace transform by a simple change of variables. If in the Mellin transform</p>
<dl>
<dd><img class="tex" alt="G(s) = \mathcal{M}\left\{g(\theta)\right\} = \int_0^\infty \theta^s g(\theta) \frac{d\theta}{\theta}" src="wikipedia-Laplace_transform_pliki/2db70a29e7768434cdd87424d52c413c.png"></dd>
</dl>
<p>we set θ = e<sup>-t</sup> we get a two-sided Laplace transform.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=14" title="Edit section: Z-transform">edit</a>]</span> <span class="mw-headline" id="Z-transform">Z-transform</span></h4>
<p>The unilateral or one-sided <a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a> is simply the Laplace transform of an ideally sampled signal with the substitution of</p>
<dl>
<dd><img class="tex" alt=" z \ \stackrel{\mathrm{def}}{=}\  e^{s T} \ " src="wikipedia-Laplace_transform_pliki/12c86e0052277a674ea7ce5ce2fc59c6.png"></dd>
</dl>
<dl>
<dd>where <img class="tex" alt="T = 1/f_s \ " src="wikipedia-Laplace_transform_pliki/33d53b1595e5164880768c03469a7f77.png"> is the <a href="http://en.wikipedia.org/wiki/Sampling_theorem" title="Sampling theorem" class="mw-redirect">sampling</a> period (in units of time e.g., seconds) and <img class="tex" alt=" f_s \ " src="wikipedia-Laplace_transform_pliki/794ff15d375e9bc879dde5dd2cf0a6b2.png"> is the <a href="http://en.wikipedia.org/wiki/Sampling_rate" title="Sampling rate">sampling rate</a> (in <a href="http://en.wikipedia.org/wiki/Sample_%28signal%29" title="Sample (signal)" class="mw-redirect">samples per second</a> or <a href="http://en.wikipedia.org/wiki/Hertz" title="Hertz">hertz</a>)</dd>
</dl>
<p>Let</p>
<dl>
<dd><img class="tex" alt=" \Delta_T(t) \ \stackrel{\mathrm{def}}{=}\  \sum_{n=0}^{\infty}  \delta(t - n T) " src="wikipedia-Laplace_transform_pliki/a7714d14ab3d65460494d9e0303304dd.png"></dd>
</dl>
<p>be a sampling impulse train (also called a <a href="http://en.wikipedia.org/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a>) and</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
x_q(t) &amp; \stackrel{\mathrm{def}}{=}\  x(t) \Delta_T(t) = x(t) \sum_{n=0}^{\infty}  \delta(t - n T) \\
&amp; = \sum_{n=0}^{\infty} x(n T) \delta(t - n T) = \sum_{n=0}^{\infty} x[n] \delta(t - n T)
\end{align}
" src="wikipedia-Laplace_transform_pliki/3119e24ec52d57b73320f2fb400e987a.png"></dd>
</dl>
<p>be the continuous-time representation of the sampled <img class="tex" alt=" x(t) \ " src="wikipedia-Laplace_transform_pliki/e34fd49d79f3869d9033f958be91021e.png"></p>
<dl>
<dd><img class="tex" alt=" x[n] \ \stackrel{\mathrm{def}}{=}\  x(nT) \ " src="wikipedia-Laplace_transform_pliki/e6b102901a66a7f80116631afc0ccbfa.png"> are the discrete samples of <img class="tex" alt=" x(t) \ " src="wikipedia-Laplace_transform_pliki/e34fd49d79f3869d9033f958be91021e.png">.</dd>
</dl>
<p>The Laplace transform of the sampled signal <img class="tex" alt=" x_q(t) \ " src="wikipedia-Laplace_transform_pliki/2b2b9bc964cc68081bf8b160b812d9e6.png"> is</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
X_q(s) &amp; = \int_{0^-}^\infty x_q(t) e^{-s t} \,dt \\
&amp; = \int_{0^-}^\infty \sum_{n=0}^\infty x[n] \delta(t - n T) e^{-s t} \, dt \\
&amp; = \sum_{n=0}^\infty x[n] \int_{0^-}^\infty \delta(t - n T) e^{-s t} \, dt \\
&amp; = \sum_{n=0}^\infty x[n] e^{-n s T}.
\end{align}
" src="wikipedia-Laplace_transform_pliki/1afacfe2d357b49a02b2373dce701d7f.png"></dd>
</dl>
<p>This is precisely the definition of the unilateral <a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a> of the discrete function <img class="tex" alt=" x[n] \ " src="wikipedia-Laplace_transform_pliki/c1466b9927640af95f78274058d272d9.png"></p>
<dl>
<dd><img class="tex" alt=" X(z) = \sum_{n=0}^{\infty} x[n] z^{-n} " src="wikipedia-Laplace_transform_pliki/93762261ebf70b67ba079c79607646fa.png"></dd>
</dl>
<p>with the substitution of <img class="tex" alt=" z \leftarrow e^{s T} \ " src="wikipedia-Laplace_transform_pliki/8a73ccf6211598a74d94187a5c966f7a.png">.</p>
<p>Comparing the last two equations, we find the relationship between the unilateral <a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a> and the Laplace transform of the sampled signal:</p>
<dl>
<dd><img class="tex" alt="X_q(s) =  X(z) \Big|_{z=e^{sT}}." src="wikipedia-Laplace_transform_pliki/20bfaeb08660a3d7aadc1fd562505c66.png"></dd>
</dl>
<p>The similarity between the Z and Laplace transforms is expanded upon in the theory of <a href="http://en.wikipedia.org/wiki/Time_scale_calculus" title="Time scale calculus">time scale calculus</a>.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=15" title="Edit section: Borel transform">edit</a>]</span> <span class="mw-headline" id="Borel_transform">Borel transform</span></h4>
<p>The integral form of the <a href="http://en.wikipedia.org/wiki/Borel_transform" title="Borel transform">Borel transform</a></p>
<dl>
<dd><img class="tex" alt="F(s) = \int_0^\infty f(z)e^{-sz}\,dz" src="wikipedia-Laplace_transform_pliki/17f04347e3595f656c56764516dd67b6.png"></dd>
</dl>
<p>is a special case of the Laplace transform for <i>ƒ</i> an <a href="http://en.wikipedia.org/wiki/Entire_function" title="Entire function">entire function</a> of <a href="http://en.wikipedia.org/wiki/Exponential_type" title="Exponential type">exponential type</a>, meaning that</p>
<dl>
<dd><img class="tex" alt="|f(z)|\le Ae^{B|z|}" src="wikipedia-Laplace_transform_pliki/9a81e6a933f9d3c32a78a449a7d1d677.png"></dd>
</dl>
<p>for some constants <i>A</i> and <i>B</i>. The generalized Borel 
transform allows a different weighting function to be used, rather than 
the exponential function, to transform functions not of exponential 
type. <a href="http://en.wikipedia.org/wiki/Nachbin%27s_theorem" title="Nachbin's theorem">Nachbin's theorem</a> gives necessary and sufficient conditions for the Borel transform to be well defined.</p>
<h4><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=16" title="Edit section: Fundamental relationships">edit</a>]</span> <span class="mw-headline" id="Fundamental_relationships">Fundamental relationships</span></h4>
<p>Since an ordinary Laplace transform can be written as a special case 
of a two-sided transform, and since the two-sided transform can be 
written as the sum of two one-sided transforms, the theory of the 
Laplace-, Fourier-, Mellin-, and Z-transforms are at bottom the same 
subject. However, a different point of view and different characteristic
 problems are associated with each of these four major integral 
transforms.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=17" title="Edit section: Table of selected Laplace transforms">edit</a>]</span> <span class="mw-headline" id="Table_of_selected_Laplace_transforms">Table of selected Laplace transforms</span></h2>
<p>The following table provides Laplace transforms for many common 
functions of a single variable. For definitions and explanations, see 
the <i>Explanatory Notes</i> at the end of the table.</p>
<p>Because the Laplace transform is a linear operator:</p>
<ul>
<li>The Laplace transform of a sum is the sum of Laplace transforms of each term.</li>
</ul>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="\mathcal{L}\left\{f(t) + g(t) \right\}  = \mathcal{L}\left\{f(t)\right\} + \mathcal{L}\left\{ g(t) \right\}  " src="wikipedia-Laplace_transform_pliki/63d29d3b6af7a937478e454f215a559c.png"></dd>
</dl>
</dd>
</dl>
<ul>
<li>The Laplace transform of a multiple of a function is that multiple times the Laplace transformation of that function.</li>
</ul>
<dl>
<dd>
<dl>
<dd><img class="tex" alt="\mathcal{L}\left\{a f(t)\right\}  = a \mathcal{L}\left\{ f(t)\right\}" src="wikipedia-Laplace_transform_pliki/9e12c0aed0cd5ccac2c44e237b64e569.png"></dd>
</dl>
</dd>
</dl>
<p>The unilateral Laplace transform takes as input a function whose time
 domain is the non-negative reals, which is why all of the time domain 
functions in the table below are multiples of the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>, u(<i>t</i>). The entries of the table that involve a time delay τ are required to be <a href="http://en.wikipedia.org/wiki/Causal_system" title="Causal system">causal</a> (meaning that τ&nbsp;&gt;&nbsp;0). A causal system is a system where the <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> <i>h</i>(<i>t</i>) is zero for all time <i>t</i> prior to <i>t</i> = 0. In general, the region of convergence for causal systems is not the same as that of <a href="http://en.wikipedia.org/wiki/Anticausal_system" title="Anticausal system">anticausal systems</a>.</p>
<table class="wikitable">
<tbody><tr>
<th>ID</th>
<th>Function</th>
<th>Time domain<br>
<img class="tex" alt="f(t) = \mathcal{L}^{-1} \left\{ F(s) \right\}" src="wikipedia-Laplace_transform_pliki/1fefd08deceab8942f35467945d2e426.png"></th>
<th>Laplace s-domain<br>
<img class="tex" alt="F(s) = \mathcal{L}\left\{ f(t) \right\}" src="wikipedia-Laplace_transform_pliki/ea58badb1eaa86d35e9659d26aa0cd49.png"></th>
<th>Region of convergence</th>
</tr>
<tr align="center">
<td>1</td>
<td>ideal delay</td>
<td><img class="tex" alt=" \delta(t-\tau) \ " src="wikipedia-Laplace_transform_pliki/d6b8857d9cceaa04f251c1593c27c381.png"></td>
<td><img class="tex" alt=" e^{-\tau s} \ " src="wikipedia-Laplace_transform_pliki/4403aea9762d86f70d7da548f0aedeb6.png"></td>
<td></td>
</tr>
<tr align="center">
<td>1a</td>
<td><a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">unit impulse</a></td>
<td><img class="tex" alt=" \delta(t) \ " src="wikipedia-Laplace_transform_pliki/c43ca1a96a98e2cfd8bf1c4b711a6fc0.png"></td>
<td><span class="texhtml">1</span></td>
<td><img class="tex" alt=" \mathrm{all} \  s \," src="wikipedia-Laplace_transform_pliki/3301cc2d426515ed6cc03f6fd4ee049a.png"></td>
</tr>
<tr align="center">
<td>2</td>
<td>delayed <i>n</i>th power<br>
with frequency shift</td>
<td><img class="tex" alt="\frac{(t-\tau)^n}{n!} e^{-\alpha (t-\tau)} \cdot u(t-\tau) " src="wikipedia-Laplace_transform_pliki/1a32acf2af69b6606c27a5e419a72961.png"></td>
<td><img class="tex" alt=" \frac{e^{-\tau s}}{(s+\alpha)^{n+1}} " src="wikipedia-Laplace_transform_pliki/85a0b518e39e0171bbc764765602f793.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; - \alpha \, " src="wikipedia-Laplace_transform_pliki/ecffaed69c2bf88a4f6f4bd093a8bd20.png"></td>
</tr>
<tr align="center">
<td>2a</td>
<td><i>n</i>th power<br>
( for integer <i>n</i> )</td>
<td><img class="tex" alt="{  t^n \over n! } \cdot u(t) " src="wikipedia-Laplace_transform_pliki/caaf716b46c9fe645c59f8b1dac7ea61.png"></td>
<td><img class="tex" alt=" { 1 \over s^{n+1} } " src="wikipedia-Laplace_transform_pliki/f63cc1917643627e2f8176704dc7824f.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"><br>
<img class="tex" alt=" (n &gt; -1) \, " src="wikipedia-Laplace_transform_pliki/4ee5e8e1a44953fbc8c99c34c8dc0c7c.png"></td>
</tr>
<tr align="center">
<td>2a.1</td>
<td><i>q</i>th power<br>
( for complex <i>q</i> )</td>
<td><img class="tex" alt="{  t^q \over \Gamma(q+1) } \cdot u(t) " src="wikipedia-Laplace_transform_pliki/82d9d628bde3f88a2355909dec62b9e8.png"></td>
<td><img class="tex" alt=" { 1 \over s^{q+1} } " src="wikipedia-Laplace_transform_pliki/de5332ba2ea3fea7c1e9c9e174a18bc9.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"><br>
<img class="tex" alt=" (\textrm{Re}\{q\} &gt; -1) \, " src="wikipedia-Laplace_transform_pliki/86da267164ef999d44d9e05d430378a7.png"></td>
</tr>
<tr align="center">
<td>2a.2</td>
<td><a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">unit step</a></td>
<td><img class="tex" alt=" u(t) \ " src="wikipedia-Laplace_transform_pliki/ff97f18dc2d82c96c88089052885e04e.png"></td>
<td><img class="tex" alt=" { 1 \over s } " src="wikipedia-Laplace_transform_pliki/914323860b4e4ab597f861de13baf25e.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>2b</td>
<td>delayed unit step</td>
<td><img class="tex" alt=" u(t-\tau) \ " src="wikipedia-Laplace_transform_pliki/4175fe50cd44b5c6be6a7e5eb43ab316.png"></td>
<td><img class="tex" alt=" { e^{-\tau s} \over s } " src="wikipedia-Laplace_transform_pliki/21238103e0ee5d6c4ee1f8baae954608.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>2c</td>
<td><a href="http://en.wikipedia.org/wiki/Ramp_function" title="Ramp function">ramp</a></td>
<td><img class="tex" alt=" t \cdot u(t)\ " src="wikipedia-Laplace_transform_pliki/9f127263906d2be1a677d0e082f10e46.png"></td>
<td><img class="tex" alt="\frac{1}{s^2}" src="wikipedia-Laplace_transform_pliki/9457e81a7d049d27d6c143c62672694e.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>2d</td>
<td><i>n</i>th power with frequency shift</td>
<td><img class="tex" alt="\frac{t^{n}}{n!}e^{-\alpha t} \cdot u(t) " src="wikipedia-Laplace_transform_pliki/16dc3a5d65d8b9ec38ae2a87acc96a48.png"></td>
<td><img class="tex" alt="\frac{1}{(s+\alpha)^{n+1}}" src="wikipedia-Laplace_transform_pliki/acdc4882eafdf7c60dd7063775fdbe05.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; - \alpha \, " src="wikipedia-Laplace_transform_pliki/ecffaed69c2bf88a4f6f4bd093a8bd20.png"></td>
</tr>
<tr align="center">
<td>2d.1</td>
<td><a href="http://en.wikipedia.org/wiki/Exponential_decay" title="Exponential decay">exponential decay</a></td>
<td><img class="tex" alt=" e^{-\alpha t} \cdot u(t)  \ " src="wikipedia-Laplace_transform_pliki/1919aaa2fef3c5ddb49df3e3471637bc.png"></td>
<td><img class="tex" alt=" { 1 \over s+\alpha } " src="wikipedia-Laplace_transform_pliki/6e880ff4aad32224561bfdd466699774.png"></td>
<td><img class="tex" alt="  \textrm{Re} \{ s \} &gt; - \alpha \ " src="wikipedia-Laplace_transform_pliki/bd17a3f11cabb0e1aab3e0d39b9c4be3.png"></td>
</tr>
<tr align="center">
<td>3</td>
<td>exponential approach</td>
<td><img class="tex" alt="( 1-e^{-\alpha t})  \cdot u(t)  \ " src="wikipedia-Laplace_transform_pliki/75041e853ab9c527869df292b5246876.png"></td>
<td><img class="tex" alt="\frac{\alpha}{s(s+\alpha)} " src="wikipedia-Laplace_transform_pliki/eaee77a3af02c3f3eede9e7065e08d81.png"></td>
<td><img class="tex" alt="  \textrm{Re} \{ s \} &gt; 0\ " src="wikipedia-Laplace_transform_pliki/7fb17272a4690432ca1ece40325a1dac.png"></td>
</tr>
<tr align="center">
<td>4</td>
<td><a href="http://en.wikipedia.org/wiki/Sine" title="Sine">sine</a></td>
<td><img class="tex" alt=" \sin(\omega t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/501044dae6527fdff868a3edc1cb146c.png"></td>
<td><img class="tex" alt=" { \omega \over s^2 + \omega^2  } " src="wikipedia-Laplace_transform_pliki/d887c67f9da1c8bd616b7cf17ab402bb.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0  \ " src="wikipedia-Laplace_transform_pliki/7fb17272a4690432ca1ece40325a1dac.png"></td>
</tr>
<tr align="center">
<td>5</td>
<td><a href="http://en.wikipedia.org/wiki/Cosine" title="Cosine" class="mw-redirect">cosine</a></td>
<td><img class="tex" alt=" \cos(\omega t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/319e74599566a5db68e42fc9e3f55533.png"></td>
<td><img class="tex" alt=" { s \over s^2 + \omega^2  } " src="wikipedia-Laplace_transform_pliki/062e1cbe552e3d25c8db6089828700bd.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \ " src="wikipedia-Laplace_transform_pliki/7fb17272a4690432ca1ece40325a1dac.png"></td>
</tr>
<tr align="center">
<td>6</td>
<td><a href="http://en.wikipedia.org/wiki/Hyperbolic_sine" title="Hyperbolic sine" class="mw-redirect">hyperbolic sine</a></td>
<td><img class="tex" alt=" \sinh(\alpha t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/110cffda17bd29af607ee867e71448cd.png"></td>
<td><img class="tex" alt=" { \alpha \over s^2 - \alpha^2 } " src="wikipedia-Laplace_transform_pliki/f534ff6109f06ff2dfae3a8260763ac9.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; | \alpha | \ " src="wikipedia-Laplace_transform_pliki/23122c906189e93db1ae3c1d64c7e1fe.png"></td>
</tr>
<tr align="center">
<td>7</td>
<td><a href="http://en.wikipedia.org/wiki/Hyperbolic_cosine" title="Hyperbolic cosine" class="mw-redirect">hyperbolic cosine</a></td>
<td><img class="tex" alt=" \cosh(\alpha t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/612a9713502dcfae76df6635778fd651.png"></td>
<td><img class="tex" alt=" { s \over s^2 - \alpha^2  } " src="wikipedia-Laplace_transform_pliki/cb2ddf7f11d671cadecb20251da912e5.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; | \alpha | \ " src="wikipedia-Laplace_transform_pliki/23122c906189e93db1ae3c1d64c7e1fe.png"></td>
</tr>
<tr align="center">
<td>8</td>
<td>Exponentially-decaying<br>
sine wave</td>
<td><img class="tex" alt="e^{-\alpha t}  \sin(\omega t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/9fe8c749e216085856c380412c460894.png"></td>
<td><img class="tex" alt=" { \omega \over (s+\alpha )^2 + \omega^2  } " src="wikipedia-Laplace_transform_pliki/aec60733f9248f3314935e649e117441.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; -\alpha \ " src="wikipedia-Laplace_transform_pliki/bd17a3f11cabb0e1aab3e0d39b9c4be3.png"></td>
</tr>
<tr align="center">
<td>9</td>
<td>Exponentially-decaying<br>
cosine wave</td>
<td><img class="tex" alt="e^{-\alpha t}  \cos(\omega t) \cdot u(t) \ " src="wikipedia-Laplace_transform_pliki/85d13ee10f823d1bb072e3ec02571f76.png"></td>
<td><img class="tex" alt=" { s+\alpha \over (s+\alpha )^2 + \omega^2  } " src="wikipedia-Laplace_transform_pliki/76ff02c32e9c46bfb0d5c93b68e47ddf.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; -\alpha \ " src="wikipedia-Laplace_transform_pliki/bd17a3f11cabb0e1aab3e0d39b9c4be3.png"></td>
</tr>
<tr align="center">
<td>10</td>
<td><i>n</i>th root</td>
<td><img class="tex" alt=" \sqrt[n]{t} \cdot u(t) " src="wikipedia-Laplace_transform_pliki/486b3056c275d0abfe2730f87a747f9f.png"></td>
<td><img class="tex" alt=" s^{-(n+1)/n} \cdot \Gamma\left(1+\frac{1}{n}\right)" src="wikipedia-Laplace_transform_pliki/56ddd75b5e2574032a977ebfc91fd817.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>11</td>
<td><a href="http://en.wikipedia.org/wiki/Natural_logarithm" title="Natural logarithm">natural logarithm</a></td>
<td><img class="tex" alt=" \ln \left (  { t \over t_0 } \right ) \cdot u(t) " src="wikipedia-Laplace_transform_pliki/60bb4bc9d7d1efc21080149a000fbe5b.png"></td>
<td><img class="tex" alt=" - { t_0 \over s}\, \left[ \ln(t_0 s)+\gamma \right] " src="wikipedia-Laplace_transform_pliki/fbf1781a263283a33cce0cc89f4b21ef.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>12</td>
<td><a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a><br>
of the first kind,<br>
of order <i>n</i></td>
<td><img class="tex" alt=" J_n( \omega t) \cdot u(t)" src="wikipedia-Laplace_transform_pliki/2d5a399929b9743c4e5a450e65326405.png"></td>
<td><img class="tex" alt="\frac{ \omega^n \left(s+\sqrt{s^2+ \omega^2}\right)^{-n}}{\sqrt{s^2 + \omega^2}}" src="wikipedia-Laplace_transform_pliki/d904648ad88d2aa2c0baa219db0c5bda.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"><br>
<img class="tex" alt=" (n &gt; -1) \, " src="wikipedia-Laplace_transform_pliki/4ee5e8e1a44953fbc8c99c34c8dc0c7c.png"></td>
</tr>
<tr align="center">
<td>13</td>
<td><a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Modified Bessel function</a><br>
of the first kind,<br>
of order <i>n</i></td>
<td><img class="tex" alt="I_n(\omega t) \cdot u(t)" src="wikipedia-Laplace_transform_pliki/39cbb0e95039d1fbf019bedacdeefc75.png"></td>
<td><img class="tex" alt=" \frac{ \omega^n \left(s+\sqrt{s^2-\omega^2}\right)^{-n}}{\sqrt{s^2-\omega^2}} " src="wikipedia-Laplace_transform_pliki/b0620cded428d2f097816c76037dd17b.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; | \omega | \, " src="wikipedia-Laplace_transform_pliki/9ea7cdf1de54b4585f4eb21ff91ab71a.png"></td>
</tr>
<tr align="center">
<td>14</td>
<td><a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a><br>
of the second kind,<br>
of order 0</td>
<td><img class="tex" alt=" Y_0(\alpha t) \cdot u(t)" src="wikipedia-Laplace_transform_pliki/fa3d7ad603bb3ccc3ca93903c64c8d29.png"></td>
<td><img class="tex" alt="-{2 \sinh^{-1}(s/\alpha) \over \pi \sqrt{s^2+\alpha^2}}" src="wikipedia-Laplace_transform_pliki/136a690f686030c12047dddebd603745.png"></td>
<td><img class="tex" alt="\textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr align="center">
<td>15</td>
<td>Modified <a href="http://en.wikipedia.org/wiki/Bessel_function" title="Bessel function">Bessel function</a><br>
of the second kind,<br>
of order 0</td>
<td><img class="tex" alt=" K_0(\alpha t) \cdot u(t)" src="wikipedia-Laplace_transform_pliki/0619012b579b2e354f5738198d027f61.png"></td>
<td>&nbsp;</td>
<td>&nbsp;</td>
</tr>
<tr align="center">
<td>16</td>
<td><a href="http://en.wikipedia.org/wiki/Error_function" title="Error function">Error function</a></td>
<td><img class="tex" alt=" \mathrm{erf}(t) \cdot u(t) " src="wikipedia-Laplace_transform_pliki/7c39eb3ecc2b14ed5c05c8c03b242de6.png"></td>
<td><img class="tex" alt="    {e^{s^2/4} \left(1 - \operatorname{erf} \left(s/2\right)\right) \over s}" src="wikipedia-Laplace_transform_pliki/4dce06ecebc971f4c8ef389ca1d8264f.png"></td>
<td><img class="tex" alt=" \textrm{Re} \{ s \} &gt; 0 \, " src="wikipedia-Laplace_transform_pliki/e1965cc834419eb9c758a02a2ff4a6a4.png"></td>
</tr>
<tr>
<td colspan="5"><b>Explanatory notes:</b>
<table class="multicol" style="background: none repeat scroll 0% 0% transparent; width: 100%;" cellpadding="0" cellspacing="0">
<tbody><tr>
<td align="left" valign="top">
<ul>
<li><img class="tex" alt=" u(t) \, " src="wikipedia-Laplace_transform_pliki/fee9613c4c57044904af0c6e64555bd4.png"> represents the <a href="http://en.wikipedia.org/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>.</li>
<li><img class="tex" alt=" \delta(t) \, " src="wikipedia-Laplace_transform_pliki/783045e83c1e5537366bd7c85dd9af27.png"> represents the <a href="http://en.wikipedia.org/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>.</li>
<li><img class="tex" alt=" \Gamma (z) \, " src="wikipedia-Laplace_transform_pliki/463c454997fe4cce591c16fa5b9ac8ee.png"> represents the <a href="http://en.wikipedia.org/wiki/Gamma_function" title="Gamma function">Gamma function</a>.</li>
<li><img class="tex" alt=" \gamma \, " src="wikipedia-Laplace_transform_pliki/44ddf6e825ef5a1ea521e708af7deb73.png"> is the <a href="http://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>.</li>
</ul>
</td>
<td align="left" valign="top">
<ul>
<li><img class="tex" alt="t \, " src="wikipedia-Laplace_transform_pliki/0c68620ee2ea4f1286fcd672a47ea080.png">, a real number, typically represents <i>time</i>,<br>
although it can represent <i>any</i> independent dimension.</li>
<li><img class="tex" alt="s \, " src="wikipedia-Laplace_transform_pliki/d0438646c1f482faffdd1bac9a841799.png"> is the <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> <a href="http://en.wikipedia.org/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, and <span class="texhtml">Re{<i>s</i>}</span> is its <a href="http://en.wikipedia.org/wiki/Real_part" title="Real part" class="mw-redirect">real part</a>.</li>
<li><img class="tex" alt=" \alpha \," src="wikipedia-Laplace_transform_pliki/b27abc434a11d07b390df859d7aa782a.png">, <img class="tex" alt=" \beta \," src="wikipedia-Laplace_transform_pliki/81b4c8dd7cbec41cae5ef37da5644e99.png">, <img class="tex" alt=" \tau \, " src="wikipedia-Laplace_transform_pliki/d95fd1519e587418ebe3da8fb081701f.png">, and <img class="tex" alt="\omega \," src="wikipedia-Laplace_transform_pliki/a70d0c9b2e529c999ec05569e1638668.png"> are <a href="http://en.wikipedia.org/wiki/Real_numbers" title="Real numbers" class="mw-redirect">real numbers</a>.</li>
<li><span class="texhtml"><i>n</i></span> is an <a href="http://en.wikipedia.org/wiki/Integer" title="Integer">integer</a>.</li>
</ul>
</td>
</tr>
</tbody></table>
</td>
</tr>
</tbody></table>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=18" title="Edit section: s-Domain equivalent circuits and impedances">edit</a>]</span> <span class="mw-headline" id="s-Domain_equivalent_circuits_and_impedances">s-Domain equivalent circuits and impedances</span></h2>
<p>The Laplace transform is often used in circuit analysis, and simple 
conversions to the s-Domain of circuit elements can be made. Circuit 
elements can be transformed into <a href="http://en.wikipedia.org/wiki/Electrical_impedance" title="Electrical impedance">impedances</a>, very similar to <a href="http://en.wikipedia.org/wiki/Phasor_%28sine_waves%29" title="Phasor (sine waves)" class="mw-redirect">phasor</a> impedances.</p>
<p>Here is a summary of equivalents:</p>
<dl>
<dd><a href="http://en.wikipedia.org/wiki/File:S-Domain_circuit_equivalency.svg" class="image"><img alt="S-Domain circuit equivalency.svg" src="wikipedia-Laplace_transform_pliki/412px-S-Domain_circuit_equivalency.png" height="427" width="412"></a></dd>
</dl>
<p>Note that the resistor is exactly the same in the time domain and the
 s-Domain. The sources are put in if there are initial conditions on the
 circuit elements. For example, if a capacitor has an initial voltage 
across it, or if the inductor has an initial current through it, the 
sources inserted in the s-Domain account for that.</p>
<p>The equivalents for current and voltage sources are simply derived from the transformations in the table above.</p>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=19" title="Edit section: Examples: How to apply the properties and theorems">edit</a>]</span> <span class="mw-headline" id="Examples:_How_to_apply_the_properties_and_theorems">Examples: How to apply the properties and theorems</span></h2>
<p>The Laplace transform is used frequently in <a href="http://en.wikipedia.org/wiki/Engineering" title="Engineering">engineering</a> and <a href="http://en.wikipedia.org/wiki/Physics" title="Physics">physics</a>; the output of a <a href="http://en.wikipedia.org/wiki/Linear_time_invariant" title="Linear time invariant" class="mw-redirect">linear time invariant</a> system can be calculated by convolving its unit <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> with the input signal. Performing this calculation in Laplace space turns the <a href="http://en.wikipedia.org/wiki/Convolution" title="Convolution">convolution</a> into a <a href="http://en.wikipedia.org/wiki/Multiplication" title="Multiplication">multiplication</a>; the latter being easier to solve because of its algebraic form. For more information, see <a href="http://en.wikipedia.org/wiki/Control_theory" title="Control theory">control theory</a>.</p>
<p>The Laplace transform can also be used to <a href="http://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations" title="Laplace transform applied to differential equations">solve differential equations</a> and is used extensively in <a href="http://en.wikipedia.org/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>. The Laplace transform reduces a linear <a href="http://en.wikipedia.org/wiki/Differential_equation" title="Differential equation">differential equation</a>
 to an algebraic equation, which can then be solved by the formal rules 
of algebra. The original differential equation can then be solved by 
applying the inverse Laplace transform. The English electrical engineer <a href="http://en.wikipedia.org/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> first proposed a similar scheme, although without using the Laplace transform; and the resulting <a href="http://en.wikipedia.org/wiki/Operational_calculus" title="Operational calculus">operational calculus</a> is credited as the Heaviside calculus.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=20" title="Edit section: Example 1: Solving a differential equation">edit</a>]</span> <span class="mw-headline" id="Example_1:_Solving_a_differential_equation">Example 1: Solving a differential equation</span></h3>
<p>In <a href="http://en.wikipedia.org/wiki/Nuclear_physics" title="Nuclear physics">nuclear physics</a>, the following fundamental relationship governs <a href="http://en.wikipedia.org/wiki/Radioactive_decay" title="Radioactive decay">radioactive decay</a>: the number of radioactive atoms <i>N</i> in a sample of a radioactive <a href="http://en.wikipedia.org/wiki/Isotope" title="Isotope">isotope</a> decays at a rate proportional to <i>N</i>. This leads to the first order linear differential equation</p>
<dl>
<dd><img class="tex" alt="\frac{dN}{dt} = -\lambda N" src="wikipedia-Laplace_transform_pliki/cb77231323dec231a3dcf672354bedec.png"></dd>
</dl>
<p>where λ is the <a href="http://en.wikipedia.org/wiki/Decay_constant" title="Decay constant" class="mw-redirect">decay constant</a>. The Laplace transform can be used to solve this equation.</p>
<p>Rearranging the equation to one side, we have</p>
<dl>
<dd><img class="tex" alt="\frac{dN}{dt} +  \lambda N  =  0. " src="wikipedia-Laplace_transform_pliki/23bd63163de96df681fdc197733a0411.png"></dd>
</dl>
<p>Next, we take the Laplace transform of both sides of the equation:</p>
<dl>
<dd><img class="tex" alt="\left( s \tilde{N}(s) - N_o  \right) + \lambda \tilde{N}(s) \ = \ 0   " src="wikipedia-Laplace_transform_pliki/3a300d2da95971f510ab4930ddd5e452.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><img class="tex" alt="\tilde{N}(s) = \mathcal{L}\{N(t)\}" src="wikipedia-Laplace_transform_pliki/740e186464727fd94a476e22bf128cd9.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="N_o \ = \ N(0)." src="wikipedia-Laplace_transform_pliki/a3bd158b63575902c64cb3b9e427a80c.png"></dd>
</dl>
<p>Solving, we find</p>
<dl>
<dd><img class="tex" alt="\tilde{N}(s) = { N_o \over s + \lambda  }." src="wikipedia-Laplace_transform_pliki/73dfebc74c2d7ca1685c2efeaaf07588.png"></dd>
</dl>
<p>Finally, we take the inverse Laplace transform to find the general solution</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
N(t) &amp; = \mathcal{L}^{-1} \{\tilde{N}(s)\} = \mathcal{L}^{-1}  \left\{ \frac{N_o}{s + \lambda} \right\} \\
&amp; = \ N_o e^{-\lambda t},
\end{align}
" src="wikipedia-Laplace_transform_pliki/3326103e58dc2742ca3d57492ecd56d0.png"></dd>
</dl>
<p>which is indeed the correct form for radioactive decay.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=21" title="Edit section: Example 2: Deriving the complex impedance for a capacitor">edit</a>]</span> <span class="mw-headline" id="Example_2:_Deriving_the_complex_impedance_for_a_capacitor">Example 2: Deriving the complex impedance for a capacitor</span></h3>
<p>In the theory of <a href="http://en.wikipedia.org/wiki/Electrical_circuit" title="Electrical circuit" class="mw-redirect">electrical circuits</a>, the current flow in a <a href="http://en.wikipedia.org/wiki/Capacitor" title="Capacitor">capacitor</a> is proportional to the capacitance and rate of change in the electrical potential (in <a href="http://en.wikipedia.org/wiki/SI" title="SI" class="mw-redirect">SI</a> units). Symbolically, this is expressed by the differential equation</p>
<dl>
<dd><img class="tex" alt=" i = C { dv \over dt} " src="wikipedia-Laplace_transform_pliki/d074af5f53c518ee607539d2018210e5.png"></dd>
</dl>
<p>where <i>C</i> is the capacitance (in <a href="http://en.wikipedia.org/wiki/Farads" title="Farads" class="mw-redirect">farads</a>) of the capacitor, <i>i</i> = <i>i</i>(<i>t</i>) is the <a href="http://en.wikipedia.org/wiki/Electric_current" title="Electric current">electric current</a> (in <a href="http://en.wikipedia.org/wiki/Amperes" title="Amperes" class="mw-redirect">amperes</a>) through the capacitor as a function of time, and <i>v</i> = <i>v</i>(<i>t</i>) is the <a href="http://en.wikipedia.org/wiki/Electrostatic_potential" title="Electrostatic potential" class="mw-redirect">voltage</a> (in <a href="http://en.wikipedia.org/wiki/Volts" title="Volts" class="mw-redirect">volts</a>) across the terminals of the capacitor, also as a function of time.</p>
<p>Taking the Laplace transform of this equation, we obtain</p>
<dl>
<dd><img class="tex" alt="  I(s) = C \left( s V(s) - V_o  \right) " src="wikipedia-Laplace_transform_pliki/a4e256a3cf904b88302ddac83c52ecaf.png"></dd>
</dl>
<p>where</p>
<dl>
<dd><img class="tex" alt=" I(s) = \mathcal{L} \{ i(t) \}, \, " src="wikipedia-Laplace_transform_pliki/ccb8e26a35234e2dbaa297e1f43e54e2.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt=" V(s) = \mathcal{L} \{ v(t) \}, \, " src="wikipedia-Laplace_transform_pliki/70389720730b160985d20a5d809cb39c.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt=" V_o \ = \ v(t)|_{t=0}. \, " src="wikipedia-Laplace_transform_pliki/22711d786c998dad18ee5009e93bf2a0.png"></dd>
</dl>
<p>Solving for <i>V</i>(<i>s</i>) we have</p>
<dl>
<dd><img class="tex" alt="  V(s) = { I(s) \over sC }  +  { V_o  \over s }.  " src="wikipedia-Laplace_transform_pliki/6c444d3842131e2d23f723210f95aa10.png"></dd>
</dl>
<p>The definition of the <a href="http://en.wikipedia.org/wiki/Complex_number" title="Complex number">complex</a> <a href="http://en.wikipedia.org/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> <i>Z</i> (in <a href="http://en.wikipedia.org/wiki/Ohm_%28unit%29" title="Ohm (unit)" class="mw-redirect">ohms</a>) is the ratio of the complex voltage <i>V</i> divided by the complex current <i>I</i> while holding the initial state <i>V</i><sub>o</sub> at zero:</p>
<dl>
<dd><img class="tex" alt="Z(s) = { V(s) \over I(s) } \bigg|_{V_o = 0}." src="wikipedia-Laplace_transform_pliki/d1aa04d9d08d562a9e95f4ae7ce0fd1c.png"></dd>
</dl>
<p>Using this definition and the previous equation, we find:</p>
<dl>
<dd><img class="tex" alt="Z(s) = \frac{1}{sC}, " src="wikipedia-Laplace_transform_pliki/60768213334c1ed61f3991d2f9c5ee98.png"></dd>
</dl>
<p>which is the correct expression for the complex impedance of a capacitor.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=22" title="Edit section: Example 3: Method of partial fraction expansion">edit</a>]</span> <span class="mw-headline" id="Example_3:_Method_of_partial_fraction_expansion">Example 3: Method of partial fraction expansion</span></h3>
<p>Consider a linear time-invariant system with <a href="http://en.wikipedia.org/wiki/Transfer_function" title="Transfer function">transfer function</a></p>
<dl>
<dd><img class="tex" alt="H(s) = \frac{1}{(s+\alpha)(s+\beta)}." src="wikipedia-Laplace_transform_pliki/4c1ace375ecbaa577d45a7acd94509c6.png"></dd>
</dl>
<p>The <a href="http://en.wikipedia.org/wiki/Impulse_response" title="Impulse response">impulse response</a> is simply the inverse Laplace transform of this transfer function:</p>
<dl>
<dd><img class="tex" alt="h(t) = \mathcal{L}^{-1}\{H(s)\}." src="wikipedia-Laplace_transform_pliki/1a6035c952d93bc851521014c8785862.png"></dd>
</dl>
<p>To evaluate this inverse transform, we begin by expanding <i>H</i>(<i>s</i>) using the method of <a href="http://en.wikipedia.org/wiki/Partial_fraction" title="Partial fraction">partial fraction expansion</a>:</p>
<dl>
<dd><img class="tex" alt="\frac{1}{(s+\alpha)(s+\beta)} = { P \over s+\alpha } + { R  \over s+\beta }." src="wikipedia-Laplace_transform_pliki/f3742d1fcc33a10d1d3f1af7f734d48d.png"></dd>
</dl>
<p>The unknown constants <i>P</i> and <i>R</i> are the <a href="http://en.wikipedia.org/wiki/Residue_%28complex_analysis%29" title="Residue (complex analysis)">residues</a> located at the corresponding <a href="http://en.wikipedia.org/wiki/Pole_%28complex_analysis%29" title="Pole (complex analysis)">poles</a> of the transfer function. Each residue represents the relative contribution of that <a href="http://en.wikipedia.org/wiki/Mathematical_singularity" title="Mathematical singularity">singularity</a> to the transfer function's overall shape. By the <a href="http://en.wikipedia.org/wiki/Residue_theorem" title="Residue theorem">residue theorem</a>, the inverse Laplace transform depends only upon the poles and their residues. To find the residue <i>P</i>, we multiply both sides of the equation by <i>s</i>&nbsp;+&nbsp;<i>α</i> to get</p>
<dl>
<dd><img class="tex" alt="\frac{1}{s+\beta} = P  + { R (s+\alpha) \over s+\beta }." src="wikipedia-Laplace_transform_pliki/d89075a11ac63488cc771af09f7711d8.png"></dd>
</dl>
<p>Then by letting <i>s</i>&nbsp;=&nbsp;−<i>α</i>, the contribution from <i>R</i> vanishes and all that is left is</p>
<dl>
<dd><img class="tex" alt="P = \left.{1 \over s+\beta}\right|_{s=-\alpha} = {1 \over \beta - \alpha}. " src="wikipedia-Laplace_transform_pliki/744ab33a67854ed29bf14edc8b01f78c.png"></dd>
</dl>
<p>Similarly, the residue <i>R</i> is given by</p>
<dl>
<dd><img class="tex" alt="R = \left.{1 \over s+\alpha}\right|_{s=-\beta} = {1 \over \alpha - \beta}." src="wikipedia-Laplace_transform_pliki/f3cc5280bee206fb77c78bbe0bc3db3f.png"></dd>
</dl>
<p>Note that</p>
<dl>
<dd><img class="tex" alt="R = {-1 \over \beta - \alpha} = - P" src="wikipedia-Laplace_transform_pliki/1e9e36efc298887fb5f271c2ac174890.png"></dd>
</dl>
<p>and so the substitution of <i>R</i> and <i>P</i> into the expanded expression for <i>H</i>(<i>s</i>) gives</p>
<dl>
<dd><img class="tex" alt="H(s)  = \left( \frac{1}{\beta-\alpha} \right) \cdot \left(  { 1 \over s+\alpha } - { 1  \over s+\beta }  \right). " src="wikipedia-Laplace_transform_pliki/1ad8272447aa3cbb2b01f621181946d3.png"></dd>
</dl>
<p>Finally, using the linearity property and the known transform for exponential decay (see <i>Item</i> #<i>3</i> in the <i>Table of Laplace Transforms</i>, above), we can take the inverse Laplace transform of <i>H</i>(<i>s</i>) to obtain:</p>
<dl>
<dd><img class="tex" alt="h(t) = \mathcal{L}^{-1}\{H(s)\} = \frac{1}{\beta-\alpha}\left(e^{-\alpha t}-e^{-\beta t}\right)," src="wikipedia-Laplace_transform_pliki/8cf1092f89a26fce3d9a820cabbf416f.png"></dd>
</dl>
<p>which is the impulse response of the system.</p>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=23" title="Edit section: Example 4: Mixing sines, cosines, and exponentials">edit</a>]</span> <span class="mw-headline" id="Example_4:_Mixing_sines.2C_cosines.2C_and_exponentials">Example 4: Mixing sines, cosines, and exponentials</span></h3>
<table class="wikitable">
<tbody><tr>
<th>Time function</th>
<th>Laplace transform</th>
</tr>
<tr>
<td><img class="tex" alt="e^{-\alpha t}\left[\cos{(\omega t)}+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}\right]u(t) " src="wikipedia-Laplace_transform_pliki/d239ce9976b7b9b57cc605fa9faf8110.png"></td>
<td><img class="tex" alt="\frac{s+\beta}{(s+\alpha)^2+\omega^2}" src="wikipedia-Laplace_transform_pliki/f22f12f0bccb7c44bb634f9c42ebd8c2.png"></td>
</tr>
</tbody></table>
<p>Starting with the Laplace transform</p>
<dl>
<dd><img class="tex" alt="X(s) = \frac{s+\beta}{(s+\alpha)^2+\omega^2}, " src="wikipedia-Laplace_transform_pliki/274373afd0838613ee1c714a72355591.png"></dd>
</dl>
<p>we find the inverse transform by first adding and subtracting the same constant α to the numerator:</p>
<dl>
<dd><img class="tex" alt="X(s) = \frac{s+\alpha } { (s+\alpha)^2+\omega^2}  +   \frac{\beta - \alpha }{(s+\alpha)^2+\omega^2}.  " src="wikipedia-Laplace_transform_pliki/b1b1ef2a1f63b77f4d78b9410b33634a.png"></dd>
</dl>
<p>By the shift-in-frequency property, we have</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
x(t) &amp; = e^{-\alpha t} \mathcal{L}^{-1} \left\{   {s \over s^2 + \omega^2}  +  {  \beta - \alpha \over s^2 + \omega^2  } \right\} \\[8pt]
&amp; = e^{-\alpha t} \mathcal{L}^{-1} \left\{   {s \over s^2 + \omega^2}  + \left( {  \beta - \alpha \over \omega } \right) \left( { \omega \over s^2 + \omega^2  } \right) \right\} \\[8pt]
&amp; = e^{-\alpha t} \left[\mathcal{L}^{-1} \left\{   {s \over s^2 + \omega^2}  \right\}  + \left( {  \beta - \alpha \over \omega } \right) \mathcal{L}^{-1} \left\{  { \omega \over s^2 + \omega^2  }  \right\}  \right].
\end{align}
" src="wikipedia-Laplace_transform_pliki/aa682ab13bda988f93ab2104c6860088.png"></dd>
</dl>
<p>Finally, using the Laplace transforms for sine and cosine (see the table, above), we have</p>
<dl>
<dd><img class="tex" alt="x(t)   =  e^{-\alpha t}  \left[\cos{(\omega t)}u(t)+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}u(t)\right]." src="wikipedia-Laplace_transform_pliki/0e1b001ae02ae6046e274b5942249705.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="x(t)   =  e^{-\alpha t}  \left[\cos{(\omega t)}+\left(\frac{\beta-\alpha}{\omega}\right)\sin{(\omega t)}\right]u(t)." src="wikipedia-Laplace_transform_pliki/4b576f7d61bdbc6852b2699a9ce54f7b.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=24" title="Edit section: Example 5: Phase delay">edit</a>]</span> <span class="mw-headline" id="Example_5:_Phase_delay">Example 5: Phase delay</span></h3>
<table class="wikitable">
<tbody><tr>
<th>Time function</th>
<th>Laplace transform</th>
</tr>
<tr>
<td><img class="tex" alt="\sin{(\omega t+\phi)} \ " src="wikipedia-Laplace_transform_pliki/2689aa995b3b60bd9a1ed9270d908268.png"></td>
<td><img class="tex" alt="\frac{s\sin\phi+\omega \cos\phi}{s^2+\omega^2} \ " src="wikipedia-Laplace_transform_pliki/d5a7715c7d3279377a30f545a95cf2dc.png"></td>
</tr>
<tr>
<td><img class="tex" alt="\cos{(\omega t+\phi)} \ " src="wikipedia-Laplace_transform_pliki/03eb304fcd6616ac38cc7fa779d97d91.png"></td>
<td><img class="tex" alt="\frac{s\cos\phi - \omega \sin\phi}{s^2+\omega^2} \ " src="wikipedia-Laplace_transform_pliki/394205125f796192956f426d1c470ef8.png"></td>
</tr>
</tbody></table>
<p>Starting with the Laplace transform,</p>
<dl>
<dd><img class="tex" alt="X(s) = \frac{s\sin\phi+\omega \cos\phi}{s^2+\omega^2}" src="wikipedia-Laplace_transform_pliki/4466106bab2c3001ad20b822aee84190.png"></dd>
</dl>
<p>we find the inverse by first rearranging terms in the fraction:</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
X(s) &amp; = \frac{s \sin \phi}{s^2 + \omega^2} + \frac{\omega \cos \phi}{s^2 + \omega^2} \\
&amp; = (\sin \phi) \left(\frac{s}{s^2 + \omega^2} \right) + (\cos \phi) \left(\frac{\omega}{s^2 + \omega^2} \right).
\end{align}
" src="wikipedia-Laplace_transform_pliki/bbbb6ea015083e4cf9e0f4a6c8f244ba.png"></dd>
</dl>
<p>We are now able to take the inverse Laplace transform of our terms:</p>
<dl>
<dd><img class="tex" alt="

\begin{align}
x(t) &amp; = (\sin \phi) \mathcal{L}^{-1}\left\{\frac{s}{s^2 + \omega^2} \right\} + (\cos \phi) \mathcal{L}^{-1}\left\{\frac{\omega}{s^2 + \omega^2} \right\} \\
&amp; =(\sin \phi)(\cos \omega t) + (\sin \omega t)(\cos \phi).
\end{align}
" src="wikipedia-Laplace_transform_pliki/36ae9d08c7f24d4135751478e9178a6e.png"></dd>
</dl>
<p>This is just the <a href="http://en.wikipedia.org/wiki/Trigonometric_identity#Angle_sum_and_difference_identities" title="Trigonometric identity" class="mw-redirect">sine of the sum</a> of the arguments, yielding:</p>
<dl>
<dd><span class="texhtml"><i>x</i>(<i>t</i>) = sin(ω<i>t</i> + φ).</span></dd>
</dl>
<p>We can apply similar logic to find that</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}^{-1} \left\{ \frac{s\cos\phi - \omega \sin\phi}{s^2+\omega^2} \right\} = \cos{(\omega t+\phi)}. \ " src="wikipedia-Laplace_transform_pliki/51aa641597d9c164698c8926ef62ff50.png"></dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=25" title="Edit section: See also">edit</a>]</span> <span class="mw-headline" id="See_also">See also</span></h2>
<ul>
<li><a href="http://en.wikipedia.org/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a></li>
<li><a href="http://en.wikipedia.org/wiki/Laplace_transform_applied_to_differential_equations" title="Laplace transform applied to differential equations">Laplace transform applied to differential equations</a></li>
<li><a href="http://en.wikipedia.org/wiki/Moment-generating_function" title="Moment-generating function">Moment-generating function</a></li>
<li><a href="http://en.wikipedia.org/wiki/Z-transform" title="Z-transform">Z-transform</a> (discrete equivalent of the Laplace transform)</li>
<li><a href="http://en.wikipedia.org/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Sumudu_transform" title="Sumudu transform">Sumudu transform</a> or <a href="http://en.wikipedia.org/wiki/Laplace%E2%80%93Carson_transform" title="Laplace–Carson transform" class="mw-redirect">Laplace–Carson transform</a></li>
<li><a href="http://en.wikipedia.org/wiki/Analog_signal_processing" title="Analog signal processing">Analog signal processing</a></li>
<li><a href="http://en.wikipedia.org/wiki/Continuous-repayment_mortgage#Ordinary_time_differential_equation" title="Continuous-repayment mortgage">Continuous-repayment mortgage</a></li>
<li><a href="http://en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_tauberian_theorem" title="Hardy–Littlewood tauberian theorem">Hardy–Littlewood tauberian theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Bernstein%27s_theorem" title="Bernstein's theorem">Bernstein's theorem</a></li>
<li><a href="http://en.wikipedia.org/wiki/Symbolic_integration" title="Symbolic integration">Symbolic integration</a></li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=26" title="Edit section: Notes">edit</a>]</span> <span class="mw-headline" id="Notes">Notes</span></h2>
<div class="reflist" style="list-style-type: decimal;">
<ol class="references">
<li id="cite_note-0"><b><a href="#cite_ref-0">^</a></b> <a href="#CITEREFKornKorn1967">Korn &amp; Korn 1967</a>, §8.1</li>
<li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <a href="#CITEREFEuler1744">Euler 1744</a>, (<a href="#CITEREFEuler1753">1753</a>) and (<a href="#CITEREFEuler1769">1769</a>)</li>
<li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <a href="#CITEREFLagrange1773">Lagrange 1773</a></li>
<li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, p.&nbsp;260</li>
<li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, p.&nbsp;261</li>
<li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, pp.&nbsp;261–262</li>
<li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <a href="#CITEREFGrattan-Guinness1997">Grattan-Guinness 1997</a>, pp.&nbsp;262–266</li>
<li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <a href="#CITEREFFeller1971">Feller 1971</a>, §XIII.1</li>
<li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <a href="#CITEREFWidder1941">Widder 1941</a>, Chapter II, §1</li>
<li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> <a href="#CITEREFWidder1941">Widder 1941</a>, Chapter VI, §2</li>
<li id="cite_note-10"><b><a href="#cite_ref-10">^</a></b> (<a href="#CITEREFKornKorn1967">Korn &amp; Korn 1967</a>, p.&nbsp;226–227)</li>
<li id="cite_note-11"><b><a href="#cite_ref-11">^</a></b> <a href="#CITEREFBracewell2000">Bracewell 2000</a>, Table 14.1, p. 385</li>
<li id="cite_note-12"><b><a href="#cite_ref-12">^</a></b> <a href="#CITEREFFeller1971">Feller 1971</a>, p.&nbsp;432</li>
</ol>
</div>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=27" title="Edit section: References">edit</a>]</span> <span class="mw-headline" id="References">References</span></h2>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=28" title="Edit section: Modern">edit</a>]</span> <span class="mw-headline" id="Modern">Modern</span></h3>
<ul>
<li><span class="citation" id="CITEREFArendtBattyHieberNeubrander2002">Arendt, Wolfgang; Batty, Charles J.K.; Hieber, Matthias; Neubrander, Frank (2002), <i>Vector-Valued Laplace Transforms and Cauchy Problems</i>, Birkhäuser Basel, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/3764365498" title="Special:BookSources/3764365498">3764365498</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Vector-Valued+Laplace+Transforms+and+Cauchy+Problems&amp;rft.aulast=Arendt&amp;rft.aufirst=Wolfgang&amp;rft.au=Arendt%2C%26%2332%3BWolfgang&amp;rft.au=Batty%2C%26%2332%3BCharles+J.K.&amp;rft.au=Hieber%2C%26%2332%3BMatthias&amp;rft.au=Neubrander%2C%26%2332%3BFrank&amp;rft.date=2002&amp;rft.pub=Birkh%C3%A4user+Basel&amp;rft.isbn=3764365498&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFBracewell2000">Bracewell, R. N. (2000), <i>The Fourier Transform and Its Applications</i> (3rd ed.), Boston: McGraw-Hill, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0071160434" title="Special:BookSources/0071160434">0071160434</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Fourier+Transform+and+Its+Applications&amp;rft.aulast=Bracewell&amp;rft.aufirst=R.+N.&amp;rft.au=Bracewell%2C%26%2332%3BR.+N.&amp;rft.date=2000&amp;rft.edition=3rd&amp;rft.place=Boston&amp;rft.pub=McGraw-Hill&amp;rft.isbn=0071160434&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFDavies2002">Davies, Brian (2002), <i>Integral transforms and their applications</i> (Third ed.), New York: Springer, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-387-95314-0" title="Special:BookSources/0-387-95314-0">0-387-95314-0</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Integral+transforms+and+their+applications&amp;rft.aulast=Davies&amp;rft.aufirst=Brian&amp;rft.au=Davies%2C%26%2332%3BBrian&amp;rft.date=2002&amp;rft.edition=Third&amp;rft.place=New+York&amp;rft.pub=Springer&amp;rft.isbn=0-387-95314-0&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFFeller1971"><a href="http://en.wikipedia.org/wiki/William_Feller" title="William Feller">Feller, William</a> (1971), <i>An introduction to probability theory and its applications. Vol. II.</i>, Second edition, New York: <a href="http://en.wikipedia.org/wiki/John_Wiley_%26_Sons" title="John Wiley &amp; Sons">John Wiley &amp; Sons</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a href="http://www.ams.org/mathscinet-getitem?mr=0270403" class="external text" rel="nofollow">0270403</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=An+introduction+to+probability+theory+and+its+applications.+Vol.+II.&amp;rft.aulast=Feller&amp;rft.aufirst=William&amp;rft.au=Feller%2C%26%2332%3BWilliam&amp;rft.date=1971&amp;rft.series=Second+edition&amp;rft.place=New+York&amp;rft.pub=%5B%5BJohn+Wiley+%26+Sons%5D%5D&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFKornKorn1967">Korn, G.A.; Korn, T.M. (1967), <i>Mathematical Handbook for Scientists and Engineers</i> (2nd ed.), McGraw-Hill Companies, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-0703-5370-0" title="Special:BookSources/0-0703-5370-0">0-0703-5370-0</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Mathematical+Handbook+for+Scientists+and+Engineers&amp;rft.aulast=Korn&amp;rft.aufirst=G.A.&amp;rft.au=Korn%2C%26%2332%3BG.A.&amp;rft.au=Korn%2C%26%2332%3BT.M.&amp;rft.date=1967&amp;rft.edition=2nd&amp;rft.pub=McGraw-Hill+Companies&amp;rft.isbn=0-0703-5370-0&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFPolyaninManzhirov1998">Polyanin, A. D.; Manzhirov, A. V. (1998), <i>Handbook of Integral Equations</i>, Boca Raton: CRC Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-8493-2876-4" title="Special:BookSources/0-8493-2876-4">0-8493-2876-4</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Handbook+of+Integral+Equations&amp;rft.aulast=Polyanin&amp;rft.aufirst=A.+D.&amp;rft.au=Polyanin%2C%26%2332%3BA.+D.&amp;rft.au=Manzhirov%2C%26%2332%3BA.+V.&amp;rft.date=1998&amp;rft.place=Boca+Raton&amp;rft.pub=CRC+Press&amp;rft.isbn=0-8493-2876-4&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFSchwartz1952">Schwartz, Laurent (1952), "Transformation de Laplace des distributions", <i>Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]</i> <b>1952</b>: 196–206, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a href="http://www.ams.org/mathscinet-getitem?mr=0052555" class="external text" rel="nofollow">0052555</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Transformation+de+Laplace+des+distributions&amp;rft.jtitle=Comm.+S%C3%A9m.+Math.+Univ.+Lund+%5BMedd.+Lunds+Univ.+Mat.+Sem.%5D&amp;rft.aulast=Schwartz&amp;rft.aufirst=Laurent&amp;rft.au=Schwartz%2C%26%2332%3BLaurent&amp;rft.date=1952&amp;rft.volume=1952&amp;rft.pages=196%E2%80%93206&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFSiebert1986">Siebert, William McC. (1986), <i>Circuits, Signals, and Systems</i>, Cambridge, Massachusetts: MIT Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-262-19229-2" title="Special:BookSources/0-262-19229-2">0-262-19229-2</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Circuits%2C+Signals%2C+and+Systems&amp;rft.aulast=Siebert&amp;rft.aufirst=William+McC.&amp;rft.au=Siebert%2C%26%2332%3BWilliam+McC.&amp;rft.date=1986&amp;rft.place=Cambridge%2C+Massachusetts&amp;rft.pub=MIT+Press&amp;rft.isbn=0-262-19229-2&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFWidder1941">Widder, David Vernon (1941), <i>The Laplace Transform</i>, Princeton Mathematical Series, v. 6, <a href="http://en.wikipedia.org/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a href="http://www.ams.org/mathscinet-getitem?mr=0005923" class="external text" rel="nofollow">0005923</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Laplace+Transform&amp;rft.aulast=Widder&amp;rft.aufirst=David+Vernon&amp;rft.au=Widder%2C%26%2332%3BDavid+Vernon&amp;rft.date=1941&amp;rft.series=Princeton+Mathematical+Series%2C+v.+6&amp;rft.pub=%5B%5BPrinceton+University+Press%5D%5D&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFWidder1945">Widder, David Vernon (1945), <a href="http://www.jstor.org/stable/2305640" class="external text" rel="nofollow">"What is the Laplace transform?"</a>, <i><a href="http://en.wikipedia.org/wiki/American_Mathematical_Monthly" title="American Mathematical Monthly">The American Mathematical Monthly</a></i> (The American Mathematical Monthly, Vol. 52, No. 8) <b>52</b> (8): 419–425, <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a href="http://dx.doi.org/10.2307%2F2305640" class="external text" rel="nofollow">10.2307/2305640</a>, <a href="http://en.wikipedia.org/wiki/International_Standard_Serial_Number" title="International Standard Serial Number">ISSN</a>&nbsp;<a href="http://www.worldcat.org/issn/0002-9890" class="external text" rel="nofollow">0002-9890</a>, <a href="http://en.wikipedia.org/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a><a href="http://www.ams.org/mathscinet-getitem?mr=0013447" class="external text" rel="nofollow">0013447</a><span class="printonly">, <a href="http://www.jstor.org/stable/2305640" class="external free" rel="nofollow">http://www.jstor.org/stable/2305640</a></span></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=What+is+the+Laplace+transform%3F&amp;rft.jtitle=%5B%5BAmerican+Mathematical+Monthly%7CThe+American+Mathematical+Monthly%5D%5D&amp;rft.aulast=Widder&amp;rft.aufirst=David+Vernon&amp;rft.au=Widder%2C%26%2332%3BDavid+Vernon&amp;rft.date=1945&amp;rft.volume=52&amp;rft.issue=8&amp;rft.pages=419%E2%80%93425&amp;rft.pub=The+American+Mathematical+Monthly%2C+Vol.+52%2C+No.+8&amp;rft_id=info:doi/10.2307%2F2305640&amp;rft.issn=0002-9890&amp;rft_id=http%3A%2F%2Fwww.jstor.org%2Fstable%2F2305640&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
</ul>
<h3><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=29" title="Edit section: Historical">edit</a>]</span> <span class="mw-headline" id="Historical">Historical</span></h3>
<ul>
<li><span class="citation" id="CITEREFDeakin1981">Deakin, M. A. B. (1981), "The development of the Laplace transform", <i>Archive for the History of the Exact Sciences</i> <b>25</b>: 343–390, <a href="http://en.wikipedia.org/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<a href="http://dx.doi.org/10.1007%2FBF01395660" class="external text" rel="nofollow">10.1007/BF01395660</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=The+development+of+the+Laplace+transform&amp;rft.jtitle=Archive+for+the+History+of+the+Exact+Sciences&amp;rft.aulast=Deakin&amp;rft.aufirst=M.+A.+B.&amp;rft.au=Deakin%2C%26%2332%3BM.+A.+B.&amp;rft.date=1981&amp;rft.volume=25&amp;rft.pages=343%E2%80%93390&amp;rft_id=info:doi/10.1007%2FBF01395660&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span></li>
</ul>
<ul>
<li><span class="citation" id="CITEREFDeakin1982">Deakin, M. A. B. (1982), "The development of the Laplace transform", <i>Archive for the History of the Exact Sciences</i> <b>26</b>: 351–381</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=The+development+of+the+Laplace+transform&amp;rft.jtitle=Archive+for+the+History+of+the+Exact+Sciences&amp;rft.aulast=Deakin&amp;rft.aufirst=M.+A.+B.&amp;rft.au=Deakin%2C%26%2332%3BM.+A.+B.&amp;rft.date=1982&amp;rft.volume=26&amp;rft.pages=351%E2%80%93381&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span></li>
</ul>
<ul>
<li><span class="citation" id="CITEREFEuler1744"><a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1744), "De constructione aequationum", <i>Opera omnia</i>, 1st series <b>22</b>: 150–161</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=De+constructione+aequationum&amp;rft.jtitle=Opera+omnia&amp;rft.aulast=Euler&amp;rft.aufirst=L.&amp;rft.au=Euler%2C%26%2332%3BL.&amp;rft.date=1744&amp;rft.series=1st+series&amp;rft.volume=22&amp;rft.pages=150%E2%80%93161&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFEuler1753"><a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1753), "Methodus aequationes differentiales", <i>Opera omnia</i>, 1st series <b>22</b>: 181–213</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Methodus+aequationes+differentiales&amp;rft.jtitle=Opera+omnia&amp;rft.aulast=Euler&amp;rft.aufirst=L.&amp;rft.au=Euler%2C%26%2332%3BL.&amp;rft.date=1753&amp;rft.series=1st+series&amp;rft.volume=22&amp;rft.pages=181%E2%80%93213&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFEuler1769"><a href="http://en.wikipedia.org/wiki/Leonhard_Euler" title="Leonhard Euler">Euler, L.</a> (1769), "Institutiones calculi integralis, Volume 2", <i>Opera omnia</i>, 1st series <b>12</b></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.atitle=Institutiones+calculi+integralis%2C+Volume+2&amp;rft.jtitle=Opera+omnia&amp;rft.aulast=Euler&amp;rft.aufirst=L.&amp;rft.au=Euler%2C%26%2332%3BL.&amp;rft.date=1769&amp;rft.series=1st+series&amp;rft.volume=12&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>, Chapters 3–5.</li>
<li><span class="citation" id="CITEREFGrattan-Guinness1997"><a href="http://en.wikipedia.org/wiki/Ivor_Grattan-Guinness" title="Ivor Grattan-Guinness">Grattan-Guinness, I</a> (1997), "Laplace's integral solutions to partial differential equations", in Gillispie, C. C., <i>Pierre Simon Laplace 1749–1827: A Life in Exact Science</i>, Princeton: Princeton University Press, <a href="http://en.wikipedia.org/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://en.wikipedia.org/wiki/Special:BookSources/0-691-01185-0" title="Special:BookSources/0-691-01185-0">0-691-01185-0</a></span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.btitle=Laplace%27s+integral+solutions+to+partial+differential+equations&amp;rft.atitle=Pierre+Simon+Laplace+1749%E2%80%931827%3A+A+Life+in+Exact+Science&amp;rft.aulast=Grattan-Guinness&amp;rft.aufirst=I&amp;rft.au=Grattan-Guinness%2C%26%2332%3BI&amp;rft.date=1997&amp;rft.place=Princeton&amp;rft.pub=Princeton+University+Press&amp;rft.isbn=0-691-01185-0&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
<li><span class="citation" id="CITEREFLagrange1773"><a href="http://en.wikipedia.org/wiki/Joseph_Louis_Lagrange" title="Joseph Louis Lagrange">Lagrange, J. L.</a> (1773), <i>Mémoire sur l'utilité de la méthode</i>, Œuvres de Lagrange, <b>2</b>, pp.&nbsp;171–234</span><span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=M%C3%A9moire+sur+l%27utilit%C3%A9+de+la+m%C3%A9thode&amp;rft.aulast=Lagrange&amp;rft.aufirst=J.+L.&amp;rft.au=Lagrange%2C%26%2332%3BJ.+L.&amp;rft.date=1773&amp;rft.series=%C5%92uvres+de+Lagrange&amp;rft.volume=2&amp;rft.pages=pp.%26nbsp%3B171%E2%80%93234&amp;rfr_id=info:sid/en.wikipedia.org:Laplace_transform"><span style="display: none;">&nbsp;</span></span>.</li>
</ul>
<h2><span class="editsection">[<a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit&amp;section=30" title="Edit section: External links">edit</a>]</span> <span class="mw-headline" id="External_links">External links</span></h2>
<table class="metadata plainlinks ambox ambox-style" style="">
<tbody><tr>
<td class="mbox-image">
<div style="width: 52px;"><img alt="" src="wikipedia-Laplace_transform_pliki/40px-Edit-clear.png" height="40" width="40"></div>
</td>
<td class="mbox-text" style="">This article's use of <a href="http://en.wikipedia.org/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a> <b>may not follow Wikipedia's <a href="http://en.wikipedia.org/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_mirror_or_a_repository_of_links.2C_images.2C_or_media_files" title="Wikipedia:What Wikipedia is not">policies</a> or <a href="http://en.wikipedia.org/wiki/Wikipedia:External_links" title="Wikipedia:External links">guidelines</a></b>. Please <a href="http://en.wikipedia.org/w/index.php?title=Laplace_transform&amp;action=edit" class="external text" rel="nofollow">improve this article</a> by removing excessive and inappropriate external links. <small><i>(November 2009)</i></small></td>
</tr>
</tbody></table>
<ul>
<li><a href="http://wims.unice.fr/wims/wims.cgi?lang=en&amp;+module=tool%2Fanalysis%2Ffourierlaplace.en" class="external text" rel="nofollow">Online Computation</a> of the transform or inverse transform, wims.unice.fr</li>
<li><a href="http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm" class="external text" rel="nofollow">Tables of Integral Transforms</a> at EqWorld: The World of Mathematical Equations.</li>
<li><span class="citation mathworld" id="Reference-Mathworld-Laplace_Transform"><a href="http://en.wikipedia.org/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, "<a href="http://mathworld.wolfram.com/LaplaceTransform.html" class="external text" rel="nofollow">Laplace Transform</a>" from <a href="http://en.wikipedia.org/wiki/MathWorld" title="MathWorld">MathWorld</a>.</span></li>
<li><a href="http://math.fullerton.edu/mathews/c2003/LaplaceTransformMod.html" class="external text" rel="nofollow">Laplace Transform Module by John H. Mathews</a></li>
<li><a href="http://fourier.eng.hmc.edu/e102/lectures/Laplace_Transform/" class="external text" rel="nofollow">Good explanations of the initial and final value theorems</a></li>
<li><a href="http://www.mathpages.com/home/kmath508/kmath508.htm" class="external text" rel="nofollow">Laplace Transforms</a> at MathPages</li>
<li><a href="http://www.intmath.com/Laplace/1a_lap_unitstepfns.php" class="external text" rel="nofollow">Laplace and Heaviside</a> at Interactive maths.</li>
<li><a href="http://www.vibrationdata.com/Laplace.htm" class="external text" rel="nofollow">Laplace Transform Table and Examples</a> at Vibrationdata.</li>
<li><a href="http://www.exampleproblems.com/wiki/index.php/PDE:Laplace_Transforms" class="external text" rel="nofollow">Examples</a> of solving boundary value problems (PDEs) with Laplace Transforms</li>
<li><a href="http://www.wolframalpha.com/input/?i=laplace+transform+example" class="external text" rel="nofollow">Computational Knowledge Engine</a> allows to easily calculate Laplace Transforms and its inverse Transform.</li>
</ul>


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